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Contents Preface vii 1. Basic Concepts 1 1.1 Thermodynamics and Heat Transfer 2 1.2 Heat Conduction 3 1.3 Convection Heat Transfer 14 1.4 Combined Conduction and Convection 17 1.5 Overall Heat Transfer Coefficient 21 1.6 Radiation Heat Transfer 23 1.7 Thermal Insulation 26 1.8 Diffusion and Mass Tran...

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Contents Preface

vii

1. Basic Concepts 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

Thermodynamics and Heat Transfer 2 Heat Conduction 3 Convection Heat Transfer 14 Combined Conduction and Convection 17 Overall Heat Transfer Coefficient 21 Radiation Heat Transfer 23 Thermal Insulation 26 Diffusion and Mass Transfer 28 Units and Dimensions 31 Solved Examples 31 Summary 44 Important Formulae and Equations 45 Review Questions 47 Objective Type Questions 48 Answers 50 Open Book Problems 50 Problems for Practice 52 References 54

2. Conduction Heat Transfer at Steady State 2.1 2.2 2.3 2.4 2.5 2.6

1

General Equation of Heat Conduction 55 Steady Heat Conduction in Simple Geometries 64 Critical Radius of Insulation 73 Extended Surfaces 75 Two- and Three-Dimensional Steady-state Heat Conduction 95 Three-Dimensional Heat Conduction 107 Solved Problems 109 Summary 144 Important Formulae and Equations 144 Objective Questions 147 Answers 150 Open Book Problems 150 Review Questions 152 Problems for Practice 153 References 157

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3. Transient Heat Conduction 3.1 3.2 3.3 3.4 3.5 3.6 3.7

158

Lumped Capacitance Method for Bodies of Infinite Thermal Conductivity 158 Plane Wall with Convection 162 Infinite Cylinder and Sphere with Convection 173 Two- and Three-Dimensional Solutions of Transient Heat Conduction 182 Semi-Infinite Solid 184 Numerical and Graphical Methods 188 Periodic Flow of Heat in one Dimension 194 Solved Problems 202 Summary 228 Important Formulae and Equations 228 Objective Type Questions 228 Answers 230 Open Book Problems 230 Review Questions 232 Problems for Practice 233 References 236

4. Convection Heat Transfer: Forced Convection 4.1 Boundary Layer Theory 237 4.2 Conservation Equations of Mass, Momentum and Energy for Laminar Flow Over a Flat Plate 245 4.3 Principle of Similarity Applied to Heat Transfer 251 4.4 Evaluation of Convection Heat Transfer Coefficients 258 4.5 Dimensional Analysis 259 4.6 Analytic Solution for Laminar Boundary Layer Flow Over a Flat Plate 269 4.7 Approximate Integral Boundary Layer Analysis 275 4.8 Turbulent Flow Over a Flat Plate: Analogy between Momentum and Heat Transfer 282 4.9 Reynolds Analogy for Turbulent Flow Over a Flat Plate 286 4.10 Constant Heat Flux Boundary Condition 288 4.11 Boundary Layer Thickness in Turbulent Flow 289 4.12 Forced Convection Inside Tubes and Ducts 294 4.13 Analysis of Laminar Forced Convection in a Long Tube 300 4.14 Analysis of Couette Flow for Laminar Forced Convection 309 4.15 Velocity Distribution in Turbulent Flow Through a Pipe 311 4.16 Analogy between Heat and Momentum Transfer in Turbulent Flow 313 4.17 Empirical Correlations 315 4.18 Flow Across a Single Circular Cylinder: Forced Convection Over Exterior Surfaces 320 4.19 Heat Transfer Enhancement 327 Solved Exampes 328 Summary 363 Important Formulae and Equations 364 Objective Type Questions 368 Answers 370 Open Book Problems 370

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Contents

xiii

Review Questions 372 Problems for Practice 375 References 380 5. Heat Transfer by Natural Convection

382

5.1 Dimensionless Parameters of Natural Convection 383 5.2 An Approximate Analysis of Laminar Natural Convection on a Vertical Plate 387 5.3 Empirical Correlations for Various Shapes 397 5.4 Rotating Cylinders, Disks and Spheres 402 5.5 Combined Forced and Natural Convection 403 Solved Examples 405 Summary 417 Important Formulae and Equations 417 Review Questions 418 Objective Type Questions 419 Answers 420 Open Book Problems 420 Problems for Practice 421 References 422 6. Condensation and Boiling 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 6.16

Dimensionless Parameters in Boiling and Condensation 424 Condensation Heat Transfer 425 Dropwise Condensation 425 Laminar Film Condensation on a Vertical Plate 426 Condensation on Horizontal Tubes 433 Condensation Number 435 Turbulent Film Condensation 436 Effect of High Vapour Velocity 436 Effect of Superheated Vapour 437 Effect of Non-Condensable Gas 437 Film Condensation Inside Horizontal Tubes 438 Boiling Heat Transfer 439 Regimes of Boiling 439 Nucleate Boiling 444 Correlations of Boiling Heat Transfer Data 446 Forced Convection Boiling 448 Solved Examples 450 Summary 458 Important Formulae and Equations 459 Review Questions 461 Objective Type Questions 462 Answers 464 Open Book Problems 464

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Problems for Practice References 466 7. Radiation Heat Transfer 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 7.13 7.14 7.15 7.16 7.17 7.18 7.19 7.20 7.21 7.22 7.23 7.24 7.25 7.26

465

468

Thermal Radiation 468 Prevost’s Theory 470 Absorptivity, Reflectivity and Transmissivity 470 Black Body 471 Emissive Power 471 Emissivity 473 Kirchhoff’s Law 474 Laboratory Black Body 476 Spectral Energy Distribution of a Black Body 477 Radiation from Real Surfaces 487 Intensity of Radiation 488 Radiant Heat Exchange between Two Black Bodies Separated by a Non Absorbing Medium 489 Shape Factor 491 Electrical Analogy 492 Radiant Heat Transfer between Two Black Surfaces Connected by Nonconducting and Reradiating Walls 494 Evaluation of Shape Factor 496 Radiation Heat Transfer Between Gray Bodies 500 Radiosity and Irradiation 502 Radiation Network for Gray Surfaces Exchanging Energy 503 Hottel’s Crossed String Method for Estimating Shape Factor for Infinitely Long Surfaces 508 Radiation Shields 510 Radiation from Cavities 513 Radiation From Gases and Vapours 514 Radiation Combined with Convection 520 Greenhouse Effect 521 Solar Radiation 521 Solved Examples 525 Summary 553 Important Formulae and Equations 554 Review Questions 556 Objective Type Questions 558 Answers 561 Open Book Problems 562 Problems for Practice 564 References 569

Contents

8. Heat Exchangers 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 8.14 8.15 8.16 8.17 8.18

Types of Heat Exchangers 570 Compact, Shell-and-Tube and Plate Heat Exchangers 572 Overall Heat Transfer Coefficient and Fouling Factor 574 Parallel Flow Heat Exchanger 576 Counter Flow Heat Exchanger 578 Use of LMTD 580 Cross-Flow Heat Exchanger 580 Comparison of Parallel Flow and Counter Flow Heat Exchangers Heat Transfer With Phase Change 583 Multipass Heat Exchangers 583 Variation of U0 along the Heating Surface 586 Effectiveness—NTU Method 587 Plate Heat Exchanger 594 Heat Transfer Enhancement 594 Heat Pipes 596 Run-around Coil Systems 604 Heat Exchanger Design Considerations 608 Selection of Heat Exchangers 608 Solved Examples 609 Summary 629 Important Formulae and Equations 630 Review Questions 631 Objective Questions 633 Answers 635 Open Book Problems 635 Problems for Practice 637 References 641

570

582

9. Some Special Heat Transfer Processes 9.1 9.2 9.3 9.4 9.5 9.6

xv

Heat Transfer in High Velocity Flows 642 Heat Transfer in Rarefied Gases 647 Transpiration and Film Cooling 653 Ablative Cooling 655 Thermodynamic Optimization of Convective Heat Transfer 656 Heat Transfer in a Circulating Fluidized Bed (CFB) Boiler 661 Solved Examples 668 Summary 673 Important Formulae and Equations 673 Review Questions 674 Objective Type Questions 675 Answers 675 Problems for Practice 675 References 676

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10. Mass Transfer 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 10.11

678

Mass Transfer By Molecular Diffusion: Fick’s Law of Diffusion 678 Equimolar Counter Diffusion 680 Molecular Diffusion Through a Stationary Gas 681 Diffusivity for Gases and Vapours 683 Concentration Boundary Layer and Mass Transfer Coefficient 684 Analogy between Momentum, Heat and Mass Transfer 685 Forced Convection Mass Transfer in Laminar Flow in a Tube 687 Mass Transfer by Convection in Turbulent Flow 688 Evaluation of Mass Transfer Coefficients by Dimensional Analysis 690 Analogy of Heat and Mass Transfer 691 Mass Transfer in Boundary Layer Flow Over a Flat Plate 692 Solved Examples 693 Summary 701 Important Formulae and Equations 701 Review Questions 703 Objective Questions 703 Answers 705 Problems for Practice 705

Appendix A: Thermophysical Properties of Matter Appendix B: Mathematical Relations and Functions Appendix C: The International System of Units Appendix D: Miscellaneous Solved Examples Appendix E: Fill in the Blanks State True or False

708 735 741 744 787 790

Bibliography Index

793 795

Basic Concepts

1

The study of transfer phenomena, which include transfer of momentum, energy, mass, electricity, etc. has been recognized as a unified discipline of fundamental importance on the basis of generalized fluxes and forces. A flux like heat transfer, momentum transfer, mass transfer, electricity and chemical reaction rate is linearly proportional to the respective conjugate force of temperature gradient, velocity gradient, concentration gradient, electric potential gradient and chemical affinity, the constant of proportionality being a property of the medium, like thermal conductivity, viscosity, diffusion coefficient and electrical conductivity. It is a law of nature (phenomenological law) which states that a driving force causes the respective flux from a higher to a lower potential. The reverse never happens spontaneously. The transfer process indicates the tendency of a system to proceed towards equilibrium. For example, in a solid body with a nonuniform temperature distribution, energy is transferred so as to establish a uniform temperature distribution in the body. Heat is defined as energy transferred by virtue of a temperature difference or gradient. Heat transfer is a vector quantity, flowing in the direction of decreasing temperature, with a negative temperature gradient. In the science of thermodynamics, the important parameter is the quantity of heat transferred during a process. In the subject of heat transfer, attention is directed to the rate at which heat is transferred. Thermodynamics is concerned with the transition of a system from one equilibrium state to another, and is based principally on the two laws of nature, the first law and the second law of thermodynamics. It is the science of heat transfer which is concerned with the estimation of the rate at which heat is transferred, the duration of heating and cooling for a certain heat duty and the surface area required to accomplish that heat duty. When a small amount of perfume vapour is sprayed into a room of air, the mass transfer process causes the perfume vapour to diffuse throughout the room until its concentration is uniform, indicating an equilibrium condition. In an electrically conducting material with a nonuniform electrical potential (voltage) distribution, electric charge will flow until a uniform potential distribution is set up. In all transfer processes we are concerned with rates at which changes in properties of a system occur. In the flow of a viscous fluid, the viscous (frictional) stresses may be related to the rate of change of momentum of a system. Heat conduction may be related to the rate of change of internal energy of system. Mass diffusion may be related to the rate of change of composition of a mixture due to transfer of one or more of the component species. There are three distinct modes in which heat transmission can take place: conduction, radiation and convection. Strictly speaking, only conduction and radiation should be classified as heat transfer processes, because only these two modes depend on the existence of a temperature difference. Convection refers to the mass motion of a fluid, and the convective heat transfer between a solid wall and a fluid depends not only on the temperature difference, but also on the mass transport of the fluid. However, since convection, like conduction and radiation, also accomplishes energy transfer from regions of higher temperature to regions of lower temperature, the term ‘heat transfer by convection’ has become generally accepted.

2

Heat and Mass Transfer

1.1 THERMODYNAMICS AND HEAT TRANSFER The science of heat transfer is concerned with the calculation of the rate at which heat flows within a medium, across an interface, or from one surface to another, and the associated temperature distribution. Thermodynamics deals with systems in equilibrium and calculates the energy transferred to change a system from one equilibrium state to another. However, it cannot tell the duration for which heat has to flow to change that state of equilibrium. For example, if 1 kg ingot of iron is quenched from 1000°C to 100°C in an oil bath, thermodynamics tells us that the loss in internal energy of the ingot is kJ ¥ 900 K = 405 kJ mcDT = 1 kg ¥ 0.45 kg K But thermodynamics cannot tell us about the time required for the temperature to drop to 100°C. The time depends on various factors such as the temperature of the oil bath, physical properties of the oil, motion of the oil etc. An appropriate heat transfer analysis considers all these factors. Analysis of heat transfer processes requires some concepts of thermodynamics. The first law of thermodynamics states the principle of conservation of energy and it is expressed in the form of an energy balance for a system. A closed system containing a fixed mass of a solid (Fig. 1.1) has a volume V(m3) and density r(kg/m3). There is heat transfer into the system at a rate Q(W), and heat may be generated internally within the Fig. 1.1 Application of first law solid, say, by nuclear fission or electrical current at a rate QG (W).The principle of energy conservation requires that over a time interval Dt(s). Change in internal energy within the system = Heat transferred into the system + Heat generated within the system DU = Q ◊ Dt + QG Dt

(1.1)

Dividing by Dt and equating it to zero in the limit. dU = Q + QG dt Now, dU = rV du = rVcdT, where du = cv dT and cv = cp = c for an incompressible fluid or a rigid solid, dT = Q + QG (1.2) rVc dt This is the energy equation on a rate basis. Figure 1.2 shows an open system for which a useful form of the first law is the steady flow energy equation (SFEE), given below: Ê ˆ Ê ˆ V2 V2 m
Á h1 + 1 + gz1 ˜ + Q
= m
Á h2 + 2 + gz2 ˜ + W
(1.3) 2 2 Ë ¯ Ë ¯ For most heat transfer equipment, changes in kinetic and potential energy are negligible and no external work is done, thus, SFEE reduces to
2
1 + Q
= mh mh

m or, Q = (h2 – h1) For an ideal gas or an incompressible liquid,

Fig. 1.2 Energy conservation for a steady flow open system

(1.4)

Basic Concepts

3

T2

h2 – h1 = ÚT cp dT 1

The second law of thermodynamics states that if two bodies at temperatures T1 and T2 are connected, and if T1 > T2, then heat will flow spontaneously and irreversibly from body 1 to body 2, causing entropy increase of the universe or entropy generation. Since, all heat transfer processes occur through finite temperature differences overcoming thermal irreversibility, the heat transfer area or operating variables can be optimized in regard to two or more irreversibilities following the principle of minimization of entropy generation or exergy destruction [3]. The roles of thermodynamics, cost or economics, and heat transfer, simultaneously act upon to yield an energy-efficient equipment, which is now a concern of the engineers.

1.2 HEAT CONDUCTION Conduction refers to the transfer of heat between two bodies or two parts of the same body through molecules which are, more or less, stationary, as in the case of solids. Fourier’s law (after the French scientist J.B.J. Fourier who proposed it in 1822) of heat conduction states that the rate of heat transfer is linearly proportional to the temperature gradient. For one-dimensional or unidirectional heat conduction dT dx dT or qk = – k (1.5) dx where qk is the rate of heat flux (a vector) in W/m2, dT/dx is the temperature gradient in the direction of heat flow x and k is the constant of proportionality, which is a property of the material through which heat propagates. This property of the material is called thermal conductivity (W/m K). The negative sign is used because heat flows from a high to a low temperature and the slope dT/dx is negative (Fig. 1.3). It may be noted that temperature can be given in kelvin or degree Celsius in Eq. (1.5) and the temperature gradient which does not depend on these units is used since one kelvin is equal to one degree Celsius (1 K = 1°C). Thus, the unit of thermal conductivity could also be written as W/m °C, but this is not the recommended practice when using the SI system of units. The magnitude of the thermal conductivity k for a given substance very much depends on its microscopic structure and also tends to vary somewhat with temperature. Table 1.1 gives some selected values of k. For the simple case of steady-state one-dimensional heat flow through a plane wall (Fig. 1.4), the temperature Fig. 1.3 Sign convention for conduction heat flow gradient and the heat flow do not vary with time, so that from Eq. (1.5) qk µ ◊

L

T2

qk Ú dx = - Ú k dT 0

T1

where the temperature at the left face (x = 0) is uniform at T1 and the temperature at the right face (x = L) is uniform at T2.

4

Heat and Mass Transfer

Table 1.1 Thermal conductivity of various materials at 0°C Material

Thermal conductivity k W/m ◊ K

Metals: Silver (pure) Copper (pure) Aluminum (pure) Nickel (pure) Iron (pure) Carbon steel, 1% C Lead (pure) Chrome-nickel steel (18% Cr, 8% Ni) Nonmetallic solids: Diamond Quartz, parallel to axis Magnesite Marble Sandstone Glass, window Maple or oak Sawdust Glass wool Ice Liquids: Mercury Water Ammonia Lubricating oil, SAE 50 Freon 12, CCl2F2 Gases: Hydrogen Helium Air Water vapor (saturated) Carbon dioxide

410 385 202 93 73 43 35 16.3 2300 41.6 4.15 2.08–2.94 1.83 0.78 0.17 0.059 0.038 2.22 8.21 0.556 0.540 0.147 0.073 0.175 0.141 0.024 0.0206 0.0146

Fig. 1.4 Temperature distribution for steady-state conduction through a plane wall and analogy between thermal and electrical circuits

Basic Concepts

If k is independent of T, we obtain after integration T -T qk = k 1 2 L If A is the surface area normal to heat flow, then the rate of heat transfer in watts is T - T2 Qk = qk A = k A 1 L Since dt/dx = – qk/k, for the same qk, if k is low (i.e. for an insulator), dT/dx will be large i.e., there will be a large temperature difference across the wall, and if k is high (i.e. for a conductor), dT/dx will be small, or there will be a small temperature difference across the wall (Fig. 1.5).

5

(1.6)

1.2.1 Resistance Concept Heat flow has an analogy in the flow of electricity. Ohm’s law states that the current I (Ampere) flowing through a wire (Fig. 1.4) is equal to the voltage potential E1 – E2 (V), divided by the electrical resistance Re (W) or E - E2 I= 1 (1.7) Re Since the temperature difference and heat flux in conduction are similar to the potential difference and electric current respectively, the rate of heat conduction through the wall [(Eq. (1.6)] can be written as Q=

T1 - T2 T1 - T2 = L / kA Rk

Fig. 1.5 Thermal resistance offered by a wall

(1.8)

where Rk = L/kA is the conductive thermal resistance to heat flow offered by the wall (Fig. 1.5). l Again, the electrical resistance Re = r , where r is the specific resistance (W m), l is the length of the A conductor (m) and A is the cross-sectional area of the conductor (m2), Eq. (1.7) can now be written as E - E2 E - E2 dE =sA 1 =sA I= 1 l l dl r A I or i = = current density (A/m2) A dE (1.9) = -s dl where s (= 1/r) is the electrical conductivity [(W m)–1 or mho] and dE/dl is the potential gradient (V/m). The similarity of Eqs (1.5) and (1.9) can be noticed. The reciprocal of the thermal resistance is referred to as the thermal conductance Kk, defined by kA (1.10) L...