OaaD
`
`ESS
`
`eedase)
`
`FUNDAMENTALS
`
`Ue Nias
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`
`WILEY
`
`JOHN WILEY & SONS
`
`New York ¢ Chichester * Brisbane ¢ Toronto ¢ Singapore
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`Dedicated to those wonderful womenin ourlives,
`
`who, through the years, have blessed us with
`their love, patience, and understanding.
`
`
`
`Copyright © 1981, 1985, 1990, by John Wiley & Sons, Inc.
`
`All rights reserved. Published simultaneously in Canada.
`
`Fundamentals of heat and mass transfer/Frank P. Incropera, David
`
`I. DeWitt, David P.,
`
`89-38319
`CIP
`
`
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`
`
`identify important dependent and
`processes and simplifying assumptions,
`independent variables, develop appropriate expressions from first principles,
`and introduce requisite material from the heat transfer knowledge base. In the
`first edition, achievement of this objective was fostered by couching manyof
`the examples and end-of-chapter problems in terms of actual engineering
`systems.
`The second edition was also driven by the foregoing objectives, as well as
`by input derived from a questionnaire sent to over 100 colleagues who used, or
`were otherwise familiar with, the first edition. A major consequence of this
`input was publication of two versions of the book, Fundamentals of Heat and
`Mass Transfer and Introduction to Heat Transfer. As in the first edition, the
`Fundamentals version included mass transfer, providing an integrated treat-
`ment of heat, mass and momentum transfer by convection and separate
`treatments of heat and mass transfer by diffusion. The Introduction version of
`the book was intended for users who embraced the treatment of heat transfer
`but did not wish to cover mass transfer effects. In both versions, significant
`improvements were made in the treatments of numerical methods and heat
`transfer with phase change.
`In this latest edition, changes have been motivated by the desire to
`expand the scope of applications and to enhance the exposition of physical
`principles. Consideration of a broader range of technically important prob-
`lems is facilitated by increased coverage of existing material on thermal
`contactresistance, fin performance, convective heat transfer enhancement, and
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`compact heat exchangers, as well as by the addition of new material on
`submerged jets (Chapter 7) and free convection in open, parallel plate chan-
`nels (Chapter 9). Submerged jets are widely used for industrial cooling and
`drying operations, while free convection in parallel plate channels is pertinent
`to passive cooling and heating systems. Expanded discussions of physical
`principles are concentrated in the chapters on single-phase convection
`(Chapters 7 to 9) and relate, for example, to forced convection in tube banks
`and to free convection on plates and in cavities. Other improvementsrelate to
`the methodology of performinga first law analysis, a more generalized lumped
`capacitance analysis, transient conduction in semi-infinjte media, and finite-
`
`the old Chapter 14, which dealt with multimode heat
`transfer problems, has been deleted and many of the problems have been
`transferred to earlier chapters. This change was motivated by recognition of
`the importance of multimode effects and the desirability of impacting student
`the earliest possible time. Hence,
`problems involving more than just a superficial consideration of multimode
`effects begin in Chapter 7 and increase in number through Chapter 13.
`in this
`the least
`important,
`improvement
`edition is the inclusion of nearly 300 new problems. In the spirit of our past
`efforts, we have attempted to address contemporary issues in many of the
`problems. Hence, as well as relating to engineering applications such as energy
`conversion and conservation, space heating and cooling, and thermal protec-
`tion, the problems deal with recent interests in electronic cooling, manufactur-
`ing, and material processing. Many of the problems are drawn from our
`accumulated research and consulting experiences;
`the solutions, which fre-
`quently are not obvious, require thoughtful implementation of the tools of heat
`transfer. It is our hope that in addition to reinforcing the student’s understand-
`ing of principles and applications, the problems serve a motivational role by
`relating the subject to real engineering needs.
`;
`Over the past nine years, we have been fortunate to have received
`constructive suggestions from many colleagues throughout the United States
`and Canada.It is with pleasure that we express our gratitude for this input.
`
`FRANK P. INCROPERA
`Davip P. DEWIrtT
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`Chapter 2.
`
`1.7. Summary
`Problems
`
`INTRODUCTION TO CONDUCTION
`2.1 The Conduction Rate Equation
`2.2 The Thermal Properties of Matter
`2.2.1 Thermal Conductivity
`2.2.2 Other Relevant Properties
`2.3. The Heat Diffusion Equation
`2.4 Boundary and Initial Conditions
`2.5 Summary
`References
`Problems
`
`Chapter 3 ONE-DIMENSIONAL, STEADY-STATE CONDUCTION
`3.1 The Plane Wall
`3.1.1 Temperature Distribution
`3.1.2 Thermal Resistance
`
`3.1.3 The Composite Wall
`3.1.4 Contact Resistance
`3.2 An Alternative Conduction Analysis
`3.3. Radial Systems
`3.3.1 The Cylinder
`3.3.2 The Sphere
`
`
`
`27
`29
`
`43
`44
`46
`47
`51
`53
`62
`65
`66
`66
`
`79
`80
`80
`82
`
`84
`86
`92
`96
`97
`103
`
`vii
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`viii©Contents
`
`3.4 Summary of One-Dimensional Conduction Results
`3.5 Conduction with Thermal Energy Generation
`
`3.5.3 Application of Resistance Concepts
`3.6 Heat Transfer from Extended Surfaces
`3.6.1 A General Conduction Analysis
`3.6.2 Fins of Uniform Cross-Sectional Area
`
`3.6.4 Overall Surface Efficiency
`3.6.5 Fin Contact Resistance
`
`J
`
`TWO-DIMENSIONAL, STEADY-STATE CONDUCTION
`
`4.2 The Method of Separation of Variables
`
`4.3.1 Methodology of Constructing a Flux Plot
`4.3.2 Determination of the Heat Transfer Rate
`4.3.3 The Conduction Shape Factor
`4.4 Finite-Difference Equations
`4.4.1 The Nodal Network
`4.4.2 Finite-Difference Form of the Heat Equation
`4.4.3 The Energy Balance Method
` Finite-Difference Solutions
`4.5.1 The Matrix Inversion Method
`4.5.2 Gauss-Seidel Iteration
`
`5.1 The Lumped Capacitance Method
`5.2 Validity of the Lumped Capacitance Method
`5.3. General Lumped Capacitance Analysis
`
`5.5. The Plane Wall with Convection
`
`5.5.2 Approximate Solution
`5.5.3 Total Energy Transfer
`5.5.4 Graphical Representations
`5.6 Radial Systems with Convection
`
`5.6.2 Approximate Solutions
`5.6.3 Total Energy Transfer
`5.6.4 Graphical Representation
`
`107
`108
`108
`114
`119
`119
`122
`123
`130
`134
`138
`141
`142
`142
`
`171
`172
`173
`177
`178
`179
`180
`184
`185
`185
`187
`194
`194
`200
`203
`203
`204
`204
`
`225
`226
`229
`234
`237
`239
`239
`240
`240
`242
`245
`245
`246
`247
`249
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`6.6 Boundary Layer Similarity: The Normalized Convection
`Transfer Equations
`6.6.1 Boundary Layer Similarity Parameters
`6.6.2 Functional Form of the Solutions
`6.7 Physical Significance of the Dimensionless Parameters
`6.8 Boundary Layer Analogics
`6.8.1 The Heat and Mass Transfer Analogy
`6.8.2 Evaporative Cooling
`6.8.3 The Reynolds Analogy
`6.9 The Effects of Turbulence
`6.10 The Convection Coefficients
`6.1} Summary
`References
`Problems
`
`Chapter 7
`
`EXTERNAL FLOW
`7.1 The Empirical Method
`7.2. The Flat Plate in Parallel Flow
`7.2.1 Laminar Flow: A Similarity Solution
`7.2.2 Turbulent Flow
`7.2.3 Mixed Boundary Layer Conditions
`7.2.4 Special Cases
`7,3. Methodology for a Convection Calculation
`
`343
`344
`346
`351
`355
`355
`359
`363
`364
`367
`368
`368
`369
`
`385
`387
`389
`389
`396
`397
`399
`401
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`7.4 The Cylinder in Cross Flow
`7.4.1 Flow Considerations
`7.4.2 Convection Heat and Mass Transfer
`
`7.6 Flow Across Banks of Tubes
`
`7.7.1 Hydrodynamic and Geometric Considerations
`7.7.2 Convection Heat and Mass Transfer
`
`’
`
`8.1 Hydrodynamic Considerations
`
`8.1.2 The Mean Velocity
`8.1.3 Velocity Profile in the Fully Developed Region
`8.1.4 Pressure Gradient and Friction Factor in Fully
`
`8.2.1 The Mean Temperature
`8.2.2 Newton’s Law of Cooling
`8.2.3 Fully Developed Conditions
`
`8.3.1 General Considerations
`8.3.2 Constant Surface Heat Flux
`8.3.3 Constant Surface Temperature
`8.4 Laminar Flow in Circular Tubes: Thermal Analysis and
`
`8.4.1 The Fully Developed Region
`
`8.5 Convection Correlations: Turbulent Flow in Circular
`
`8.6 Convection Correlations: Noncircular Tubes
`8.7 The Concentric Tube Annulus
`8.8 Heat Transfer Enhancement
`8.9 Convection Mass Transfer
`
`9.2. The Governing Equations
`
`9.4 Laminar Free Convection on a Vertical Surface
`9.5 The Effects of Turbulence
`
`408
`408
`411
`417
`420
`431
`431
`433
`438
`440
`441
`442
`
`467
`468
`468
`469
`470
`
`472
`474
`475
`476
`476
`480
`480
`482
`485
`
`489
`489
`494
`
`495
`501
`502
`504
`505
`507
`509
`510
`
`529
`530
`533
`535
`536
`539
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`10.4.1 Nucleate Pool Boiling
`10.4.2 Critical Heat Flux for Nucleate Pool Boiling
`10.4.3 Minimum Heat Flux
`10.4.4 Film Pool Boiling
`10.4.5 Parametric Effects on Pool Boiling
`10.5 Forced-Convection Boiling
`10.5.1 External Forced-Convection Boiling
`10.5.2 Two-Phase Flow
`10.6 Condensation: Physical Mechanisms
`10.7. Laminar Film Condensation on a Vertical Plate
`10.8 Turbulent Film Condensation
`10.9 Film Condensation on Radial Systems
`10.10 Film Condensation in Horizontal Tubes
`10.11 Dropwise Condensation
`10.12 Summary
`References
`Problems
`
`Chapter 11 HEAT EXCHANGERS
`11.1 Heat Exchanger Types
`11.2 The Overall Heat Transfer Coefficient
`11.3 Heat Exchanger Analysis: Use of the Log Mean
`Temperature Difference
`11.3.1 The Parallel-Flow Heat Exchanger
`
`596
`597
`598
`599
`600
`606
`606
`607
`608
`610
`615
`619
`622
`623
`624
`624
`627
`
`639
`640
`642
`
`645
`646
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`11.3.2 The Counterflow Heat Exchanger
`11.3.3 Special Operating Conditions
`11.3.4 Multipass and Cross-Flow Heat Exchangers
`11.4 Heat Exchanger Analysis: The Effectiveness-NTU
`
`11.4.2 Effectiveness-NTU Relations
`11.5 Methodology of a Heat Exchanger Calculation
`11.6 Compact Heat Exchangers
`
`RADIATION: PROCESSES AND PROPERTIES
`
`12.2.2 Relation to Emission
`12.2.3 Relation to Irradiation
`12.2.4 Relation to Radiosity
`
`12.3.1 The Planck Distribution
`12.3.2 Wien’s Displacement Law
`12.3.3 The Stefan-Boltzmann Law
`
`12.5 Surface Absorption, Reflection, and Transmission
`
`12.5.4 Special Considerations
`
`RADIATION EXCHANGE BETWEEN SURFACES
`
`13.1.1 The View Factor Integral
`13.1.2 View Factor Relations
`13.2 Blackbody Radiation Exchange
`13.3 Radiation Exchange Between Diffuse, Gray Surfaces
`
`13.3.1 Net Radiation Exchange at a Surface
`13.3.2 Radiation Exchange Between Surfaces
`13.3.3 The Two-Surface Enclosure
`
`13.3.5 The Reradiating Surface
`
`649
`650
`650
`658
`658
`660
`666
`672
`678
`679
`680
`
`695
`696
`699
`699
`702
`706
`708
`709
`710
`712
`712
`713
`719
`729
`731
`732
`734
`734
`740
`742
`749
`7356
`758
`759
`
`791
`792
`792
`794
`803
`
`806
`806
`808
`814
`816
`819
`
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`14.4.4 Evaporation in a Column
`14.5 Mass Diffusion with Homogeneous Chemical Reactions
`14.6 Transient Diffusion
`References
`Problems
`
`Appendix A
`
`THERMOPHYSICAL PROPERTIES OF MATTER
`
`MATHEMATICAL RELATIONS AND FUNCTIONS
`
`AN INTEGRAL LAMINAR BOUNDARY LAYER
`SOLUTION FOR PARALLEL FLOW OVER A FLAT
`PLATE
`
`Appendix B
`
`Appendix C
`
`Index
`
`900
`902
`906
`910
`911
`
`AL
`
`Bl
`
`Cl
`
`Il
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`SYMBOLS
`
`Gz
`
`Gractz number
`
`hyp
`hy
`Arad
`
`J*
`
`gravitational acceleration, m/s*
`gravitational constant,
`lkg-m/N-s? or
`32.17 ft + Ib,,/Ib, - s
`nozzle height, m
`convection heat transfer coefficient,
`W/m? - K; Planck’s constant
`latent heat of vaporization, J/kg
`convection masstransfer coefficient, m/s
`radiation heat transfer coefficient,
`W/m? -K
`electric current, A; radiation intensity,
`W/m? - sr
`electric current density, A/m?; enthalpy
`per unit mass, J/kg
`radiosity, W/m?
`Jakob number
`diffusive molar flux of species i relative
`to the mixture molar average velocity,
`kmol/s - m2
`diffusive massflux of species i relative to
`the mixture mass average velocity,
`kg/s - m?
`Colburn j factor for heat transfer
`Colburn / factor for mass transfer
`thermal conductivity, W/m - K;
`Boltzmann’s constant
`zero-order, homogeneousreaction rate
`constant, kmol/s - m3
`first-order, homogeneousreaction rate
`constant, 5
`!
`first-order, homogeneousreaction rate
`constant, m/s
`characteristic length, m
`Lewis number
`mass, kg; numberof heat transfer lanes
`in a flux plot; reciprocal of the Fourier
`numberfor finite-difference solutions
`rate of transfer of mass for species i,
`kg/s
`rate of increase of massof species i duc
`to chemical reactions, kg/s
`tate at which mass enters a control
`volume, kg/s
`
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`
`numberof tubesin longitudinal and
`transverse directions
`dimensionless longitudinal and transverse
`pitch of a tube bank
`perimeter, m; general fluid property
`designation
`Peclet number ( RePr)
`Prandtl number
`pressure, N/m?
`encrgy transfer, J
`heat transfer rate, W
`rate of energy generation per unit volume,
`W/m?
`:
`heat transfer raté per unit length, W/m
`heat flux, W/m?
`cylinder radius, m
`universal gas constant
`Rayleigh number
`Reynolds number
`electric resistance, Q
`fouling factor, m? -K/W
`mass transfer resistance, s/m?
`residual for the m,n nodal point
`thermal resistance, K/W
`thermal contact resistance, K/W
`cylinder or sphere radius, m
`
`turbulence, m
`concentration entry length, m
`hydrodynamic entry length, m
`thermal entry length, m
`mole fraction of species i, C,/C
`
`Xtde
`Xida
`Mat
`x;
`
`Greek Letters
`a
`thermal diffusivity, m?/s; heat exchanger
`surface area per unit volume, m?/m?;
`absorptivity
`volumetric thermal expansion coefficient,
`K-}
`mass flow rate per unit width in film
`condensation, kg/s -m
`hydrodynamic boundary layer thick-
`ness, m
`concentration boundary layer thick-
`ness, m
`thermal boundary layer thickness, m
`emissivity; porosity of a packed bed; heat
`exchanger effectiveness
`fin effectiveness
`turbulent diffusivity for heat transfer,
`m?/s
`turbulent diffusivity for momentum
`transfer, m?/s
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`fully developed conditions
`saturated vapor conditions
`heat transfer conditions
`hydrodynamic; hot fluid
`general species designation; inner surface
`of an annulus; initial condition; tube
`inlet condition; incident radiation
`based on characteristic length
`saturated liquid conditions
`log mean condition
`momentum transfer condition
`mass transfer condi tion; mean value over
`a tube cross section
`maximum fluid velocity
`center or midplane condition; tube outlet
`condition; outer
`reradiating surface
`reflected radiation
`radiation
`solar conditions
`surface conditions; solid properties
`saturated conditions
`sky conditions
`surroundings
`thermal
`transmitted
`saturated vapor conditions
`local conditions on a surface
`spectral
`free stream conditions
`
`max
`
`r,ref
`rad
`
`sky
`sur
`
`Superscripts
`fluctuating quantity
`molar average; dimensionless quantity
`
`Overbar
`
`surface average conditions; time mean
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`
`
`
`INTRODUCTION
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`From the study of thermodynamics, you have learned that energy can be
`transferred by interactions of a system with its surroundings. These interac-
`tions are called work and heat. However, thermodynamics deals with the end
`states of the process during which an interaction occurs and provides no
`information concerning the nature of the interaction or the time rate at which
`is to extend thermodynamic analysis
`transfer and through development of
`relations to calculate heat transfer raies. In this chapter we lay the foundation
`for much of the material treated in the text. We do so by raising several
`questions. What is heat transfer? Howis heat transferred? Whyis it important to
`studyit? In answering these questions, we will begin to appreciate the physical
`mechanisms that underlie heat transfer processes and the relevance of these
`processes to our industrial and environmental problems.
`
`A simple, yet general, definition provides sufficient response to the question:
`
`Heat transfer (or heat) is energy in transit due to a temperature difference.
`Wheneverthere exists a temperature difference in a medium or between media,
`
`transfer
`types of heat
`As shown in Figure 1.1, we refer to different
`processes as modes. When a temperature gradient exists in a stationary
`medium, which may be a solid or a fluid, we use the term conduction to refer to
`transfer that will occur across the medium. In contrast,
`the term
`convection tefers to heat transfer that will occur between a surface and a
`moving fluid when they are at different temperatures. The third modeof heat
`transfer is termed thermal radiation. All surfaces of finite temperature emit
`energy in the form of electromagnetic waves. Hence, in the absence of an
`transfer by radiation between two
`
`Convection from a surface
`to a movingfluid
`
`Net radiation heat exchange
`between two surfaces
`
`Ts > Teo
`Movingfluid, T'.,
`q”
`
`7.
`
`Surface, T
`L
`St
`Ys
`a
`l\s Surface, Ts
`“mw
`
`Figure 1.1 Conduction, convection, and radiation heat transfer modes.
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`
`higher molecular energies, and when neighboring moleculescollide, as they are
`constantly doing, a transfer of energy from the more energetic to the less
`energetic molecules must occur. In the presence of a temperature gradient,
`energy transfer by conduction must then occur in the direction of decreasing
`temperature. This transfer is evident from Figure 1.2. The hypothetical plane
`at x, is constantly being crossed by molecules from above and below due to
`their random motion. However, molecules from above are associated with a
`larger temperature than those from below, in which case there must be a nef
`
`' [aanoe
`Q 7 Path’
`
`
`
`Figure 1.2 Association of conduction heat transfer with diffusion of energy due to
`molecular activity.
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`transfer of energy in the positive x direction. We may speak of the net transfer
`of energy by random molecular motion as a diffusion of energy.
`The situation is much the same in liquids, although the molecules are
`more closely spaced and the molecular interactions are stronger and more
`frequent. Similarly, in a solid, conduction may be attributed to atomic activity
`in the form of lattice vibrations. The modern view is to ascribe the energy
`transfer to lattice waves induced by atomic motion. In a nonconductor,
`the
`energy transfer is exclusively via these lattice waves; in a conductorit is also
`due to the translational motion of the free electrons. We treat the important
`properties associated with conduction phenomena in Chapter 2 and in Ap-
`
`transfer are legion. The exposed end of a
`metal spoon suddenly immersed in a cup of hot coffee will eventually be
`warmed due to the conduction of energy through the spoon. On a winter day
`there is significant energy loss from a heated room to the outside air. This loss
`is principally due to conduction heat transfer through the wall that separates
`
`It is possible to quantify heat transfer processes in terms of appropriate
`rate equations. These equations may be used to compute the amountof energy
`time, For heat conduction,
`the rate equation is
`known as Fourier’s law. For the one-dimensional plane wall shown in Figure
`1.3, having a temperature distribution T(x), the rate equation is expressed as
`
`;
`(1.1)
`
`The heat flux gq‘ (W/m?) is the heat transfer rate in the x direction per unit
`area perpendicular to the direction of transfer, and it is proportional to the
`temperature gradient, d7'/dx, in this direction. The proportionality constant k
`is a transport property known as the thermal conductivity (W/m - K) and is a
`characteristic of the wall material. The minussign is a consequenceof the fact
`that heat is transferred in the direction of decreasing temperature. Under the
`steady-state conditions shown in Figure 1.3, where the temperature distribu-
`
`(diffusion of energy).
`
`Figure 1.3 One-dimensional heat transfer by conduction
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`inner and outer surfaces, respectively. What is the rate of heat loss through a
`wall which is 0.5 m by 3 m onaside?
`
`
`
`SOLUTION
`
`Find: Wall heatloss.
`
`Schematic:
`
`Known: Steady-state conditions with prescribed wall thickness, area, ther-
`mal conductivity, and surface temperatures.
`
`Ty = 1400 K-
`
`T= 1150 K—
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`2. One-dimensional conduction through the wall.
`
`Analysis: Since heat transfer through the wall is by conduction, the heat
`flux may be determined from Fourier’s law. Using Equation 1.2
`
`
`250 K
`
`0.15 m
`
`= 2833 W/m?
`
`The heat flux represents the rate of heat transfer through a section of unit
`
`q, = (AW) qv = (0.5 m X 3.0 m) 2833 W/m? = 4250 W
`
`<J
`
`2. Note distinction between heat flux and heatrate.
`
`In
`two mechanisms.
`transfer mode is comprised of
`addition to energy transfer due to random molecular motion (diffusion), there is
`also energy being transferred by the bulk, or macroscopic, motion of the fluid.
`This fluid motionis associated with the fact that, at any instant, large numbers
`of molecules are moving collectively or as aggregates. Such motion, in the
`presence of a temperature gradient, will give rise to heat transfer. Because the
`molecules in the aggregate retain their random motion,the total heat transfer
`is then due to a superposition of energy transport by the random motion of
`the molecules and by the bulk motion of the fluid. It is customary to use the
`term convection when referring to this cumulative transport and the term
`advection when referring to transport due to bulk fluid motion.
`Weare especially interested in convection heat transfer, which occurs
`between a fluid in motion and a bounding surface when the two are at
`different temperatures. Consider fluid flow over the heated surface of Figure
`1.4. A consequence of the fluid—surface interaction is the development of a
`region in thefluid through which the velocity varies from zero at the surface to
`a finite value u,, associated with the flow. This region of the fluid is known as
`the hydrodynamic, or velocity, boundary layer. Moreover, if the surface and
`flow temperatures differ, there will be a region of the fluid through which the
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`from the fact that the boundary layer grows as the flow progresses in the x
`direction.
`In effect,
`the heat
`that
`is conducted into this layer
`is swept
`downstream and is eventually transferred to the fluid outside the boundary
`layer. Appreciation of boundary layer phenomenais essential to understand-
`ing convection heat transfer. It is for this reason that the discipline of fluid
`mechanics plays a vital role in our later analysis of convection.
`Convection heat transfer may be classified according to the nature of the
`flow. We speak of forced convection when the flow is caused by external
`means, such as by a fan, a pump, or atmospheric winds. As an example,
`consider the use of a fan to provide forced convection air cooling of hot
`electrical components on a stack of printed circuit boards (Figure 1.5a). In
`contrast, for free (or natural) convection the flow is induced by buoyancy
`forces whicharise from density differences caused by temperaturevariations in
`the fluid. An example is the free convection heat transfer that occurs from hot
`components on a vertical array of circuit boardsin still air (Figure 1.5b). Air
`that makes contact with the components experiences an increase in tempera-
`ture and hence a reduction in density. Since it
`is now lighter than the
`surrounding air, buoyancy forces induce a vertical motion for which warm air
`ascending from the boardsis replaced by an inflow of cooler ambientair.It is
`useful to note that, while we have presumed pure forced convection in Figure
`15a and pure natural convection in Figure 1.55, conditions corresponding to
`mixed (combined) forced and natural convection may exist. For example, if
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`Buoyancy-driven 4s
`flow
`[
`Hot components—
`on printed
`circuit boards
`
`_]
`
`q”
`
`||
`
`||
`
`|
`|
`|
`
`Pi t t
`
`Moist air
`q”
`
`(b)
`
`Cold
`water
`
` Hotplate
`
`Figure 1.5 Convection heat transfer processes. (a) Forced convection. (b) Natural
`
`(d)
`
`velocities associated with the flow of Figure 1.5a are small and/or buoyancy
`forces are large, a secondary flow that is comparable to the imposed forced
`flow could be induced. The buoyancy induced flow would be normal to the
`forced flow and would haveasignificant effect on convection heat transfer
`from the components. In Figure 1.55 mixed convection would result if a fan
`were used to force air upward through the circuit boards, thereby assisting the
`buoyancy flow, or downward, thereby opposing the buoyancyflow.
`We have described the convection heat transfer mode as energy transfer
`occurring within a fluid due to the combinedeffects of conduction and bulk
`fluid motion. Typically, the energy that is being transferred is the sensible, or
`internal thermal, energy of the fluid. However, there are convection processes
`for which thereis, in addition, /atent heat exchange. This latent heat exchange
`is generally associated with a phase change betweenthe liquid and vaporstates
`of the fluid. Two special cases of interest
`in this text are boiling and
`condensation. For example, convection heat transfer results from fluid motion
`induced by vapor bubbles generated at the bottom of a pan of boiling water
`(Figure 1.5c) or by the condensation of water vapor on the outer surface of a
`
`Regardless of the particular nature of the convection heat transfer pro-
`cess, the appropriate rate equation is of the form
`
`(1.3a)
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`the means by which 4 may be determined. Although consideration of these
`meansis deferred to Chapter 6, convection heat transfer will frequently appear
`as a boundary condition in the solution of conduction problems (Chapters 2 to
`5). In the solution of such problems we presume h to be known,using typical
`values given in Table 1.1.
`When Equation 1.3a is used, the convection heat flux is presumed to be
`positive if heat is transferred from the surface (7, > T,,) and negative if heat
`is transferred to the surface (7,, > T,). However, if 7, > T,, there is nothing
`which precludes us from expressing Newton’s law of cooling as
`
`q” = KT, - T,)
`
`(1:3)
`
`in which case heat transfer is positive if it is to the surface.
`
`1.2.3 Radiation
`
`Thermal radiation is energy emitted by matter that is at a finite temperature.
`Although we focus primarily on radiation from solid surfaces, emission may
`also occur from liquids and gases. Regardless of the form of matter,
`the
`emission may be attributed to changes in the electron configurations of the
`constituent atoms or molecules. The energy of the radiation field is trans-
`ported by electromagnetic waves (or alternatively, photons). While the transfer
`of energy by conduction or convection requires the presence of a material
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`10=Chapter 1=Introduction
`
`medium, radiation does not. In fact, radiation transfer occurs mostefficiently
`
`The maximum flux (W/m) at which radiation may be emitted from a
`surface is given by the Stefan—Boltzmann law
`
`(1.4)
`
`the surface and o is the
`the absolute temperature (K) of
`Stefan—Boltzmann constant (o = 5.67 X 107® W/m? - K*). Such a surfaceis
`called an ideal radiator or blackbody. The heat flux emitted by a real surface is
`less than that of the ideal radiator and is given by
`°
`
`1.5
`
`is a radiative property of the surface called the emissivity. This
`property, whose value is in the range 0 < « < 1, indicates howefficiently the
`surface emits compared to an ideal radiator. Conversely,
`if radiation is
`incident upon a surface, a portion will be absorbed, and the rate at which
`energy is absorbed per unit surface area may be evaluated from knowledge of
`a surface radiative property termed the absorptivity a. That1s,
`
`(1.6)
`
`where 0 < a < 1. Whereas radiation emission reduces the thermal energy of
`
`Equations 1.5 and 1.6 determine the rate at which radiant energy is
`emitted and absorbed, respectively, at a surface. Determination of the ne? rate
`at which radiation is exchanged between surfaces is generally a good deal more
`complicated. However, a special case that occurs frequently in practice in-
`volves the net exchange between a small surface and a much larger surface
`that completely surrounds the smaller one (Figure 1.6). The surface and the
`surroundings are separated by a gas that has no effect on the radiation
`transfer. Assuming the surface to be one for which a = e (a gray surface), the
`
`
`
`Convection
`heattransfer
`
`
`
`Figure 1.6 Radiation exchange between a surface and its
`
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`convection heat transfer coefficient h is generally weak.
`The surface within the surroundings may also simultaneously transfer
`heat by convection to the adjoining gas (Figure 1.6). The total rate of heat
`transfer from the surface is then the sum of the heat rates due to the two
`modes. Thatis,
`
`q= Qeonv t Grad
`
`or,
`
`q = hA(T, — T,.) + eAo(Ti - Ti.)
`
`(1.10)
`
`Note that the convection heat transfer rate q,,,. is simply the product of the
`flux given by Equation 1.3a and the surface area.
`
`EXAMPLE1.2
`
`An uninsulated steam pipe passes through a room in which theair and walls
`are at 25°C. The outside diameter of the pipe is 70 mm, and its surface
`temperature and emissivity are 200°C and 0.8, respectively. If the coefficient
`associated with free convection heat transfer from the surface to the air is
`15 W/m’- K, whatis the rate of heat loss from the surface per unit length of
`pipe?
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`
`
`Known: Uninsulated pipe of prescribed diameter, emissivity, and surface
`temperature in a room with fixed wall and air temperatures.
`
`Find: Pipe heat loss per unit length, q’ (W/m).
`
`
`
`2. Radiation exchange between the pipe and the room is between a small
`surface enclosed within a muchlarger surface.
`
`Analysis: Heat loss from the pipe is by convection to the room air and by
`radiation exchange with the walls. Hence, from Equation 1.10, with A =
`
`q =h(aDL)(T, — T,,) + e(mDL)o(T; — TS.)
`
`The heat loss per unit length of pipe is then
`q’= - = 15 W/m- K (7 x 0.07 m)(200 — 25)°C
`
`+0.8(m x 0.07 m) 5.67 x 107° W/m? - K4 (4734 — 2984) K4
`
`q’ = 577 W/m + 421 W/m = 998 W/m
`
`4
`
`
`
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`
`second laws, it considers neither the mechanisms that provide for heat ex-
`change nor the methods that exist for computing the rate of heat exchange.
`Thermodynamics is concerned with equilibrium states of matter, where an
`equilibrium state necessarily precludes the existence of a temperature gradient.
`Although thermodynamics may be used to determine the amount of energy
`required in the form of heat for a system to pass from one equilibrium state to
`another, it does not acknowledge that heat transfer is inherently a nonequilib-
`rium process. Forheat transfer to occur, there must be a temperature gradient,
`hence thermodynamic nonequilibrium. The discipline of heat transfer there-
`fore seeks to do what thermodynamics is inherently unable to do.It seeks to
`quantify the rate at which heat transfer occurs in terms of the degree of
`thermal nonequilibrium. This is done through the rate equations for the three
`modes, expressed by Equations 1.1, 1.3, and 1.7.
`
`1.3
`
`THE CONSERVATION OF ENERGY REQUIREMENT
`
`The subjects of thermodynamics and heat transfer are highly complementary.
`For example, heat
`transfer is an extension of thermodynamics in that
`it
`considers the rate at which energ