Introduction to Convective Heat Transfer

Nevzat Onur is Emeritus Professor of Mechanical Engineering at Gazi University. He pursued his undergraduate studies in mechanical engineering at the University of California, Davis, USA where he received his B.S. degree in 1974. He then attended the Tennessee Technological University, Cookeville, USA completing his M.S. and Ph.D. degree in 1976 and 1980. He taught at different universities in Turkey and he retired from Gazi University in 2011. He has over thirty years’ experience in heat transfer research and development. His research interests have mainly been in viscous flow and convection heat transfer. He lives in Ankara, Turkey.

Preface xv

About the Author xvii

About the Companion Website xviii

1 Foundations of Convective Heat Transfer 1

1.1 Fundamental Concepts 1

1.2 Coordinate Systems 1

1.3 The Continuum and Thermodynamic Equilibrium Concepts 2

1.4 Velocity and Acceleration 3

1.5 Description of a Fluid Motion: Eulerian and Lagrangian Coordinates and Substantial Derivative 4

1.5.1 Lagrangian Approach 4

1.5.2 Eulerian Approach 5

1.6 Substantial Derivative 7

1.7 Conduction Heat Transfer 10

1.8 Fluid Flow and Heat Transfer 11

1.9 External Flow 11

1.9.1 Velocity Boundary Layer and Newton’s Viscosity Relation 11

1.9.2 Thermal Boundary Layer 12

1.10 Internal Flow 19

1.10.1 Mean Velocity 19

1.10.2 Mean Temperature 20

1.11 Thermal Radiation Heat Transfer 22

1.12 The Reynolds Transport Theorem: Time Rate of Change of an Extensive Property of a System Expressed in Terms of a Fixed Finite Control Volume 22

Problems 28

References 31

2 Fundamental Equations of Laminar Convective Heat Transfer 33

2.1 Introduction 33

2.2 Integral Formulation 33

2.2.1 Conservation of Mass in Integral Form 33

2.2.2 Conservation of Linear Momentum in Integral Form 34

2.2.3 Conservation of Energy in Integral Form 36

2.3 Differential Formulation of Conservation Equations 38

2.3.1 Conservation of Mass in Differential Form 38

2.3.1.1 Cylindrical Coordinates 41

2.3.1.2 Spherical Coordinates 41

2.3.2 Conservation of Linear Momentum in Differential Form 42

2.3.2.1 Equation of Motion for a Newtonian Fluid with Constant Dynamic Viscosity μ and Density ρ 45

2.3.2.2 Cartesian Coordinates (x, y, z) 45

2.3.2.3 Cylindrical Coordinates (r, θ,z) 46

2.3.2.4 Spherical Coordinates (r, θ, ϕ) 46

2.3.3 Conservation of Energy in Differential Form 47

2.3.3.1 Mechanical Energy Equation 53

2.3.3.2 Thermal Energy Equation 53

2.3.3.3 Thermal Energy Equation in Terms of Internal Energy 54

2.3.3.4 Thermal Energy Equation in Terms of Enthalpy 55

2.3.3.5 Temperature T and Constant Volume Specific Heat CV 55

2.3.3.6 Temperature and Constant Pressure Specific Heat cp 56

2.3.3.7 Special Cases of the Differential Energy Equation 58

2.3.3.8 Perfect Gas and the Thermal Energy Equation Involving T and cp 58

2.3.3.9 Perfect Gas and the Thermal Energy Equation Involving T and CV 58

2.3.3.10 An Incompressible Pure Substance 58

2.3.3.11 Rectangular Coordinates 59

2.3.3.12 Cylindrical Coordinates (r, θ, z) 59

2.3.3.13 Spherical Coordinates (r, θ, ϕ) 59

Problems 64

References 67

3 Equations of Incompressible External Laminar Boundary Layers 69

3.1 Introduction 69

3.2 Laminar Momentum Transfer 69

3.3 The Momentum Boundary Layer Concept 70

3.3.1 Scaling of Momentum Equation 71

3.4 The Thermal Boundary Layer Concept 76

3.4.1 Scaling of Energy Equation 77

3.5 Summary of Boundary Layer Equations of Steady Laminar Flow 82

Problems 82

References 83

4 Integral Methods in Convective Heat Transfer 85

4.1 Introduction 85

4.2 Conservation of Mass 85

4.3 The Momentum Integral Equation 87

4.3.1 The Displacement Thickness δ1 88

4.3.2 Momentum Thickness δ2 89

4.4 Alternative Form of the Momentum Integral Equation 90

4.5 Momentum Integral Equation for Two-Dimensional Flow 90

4.6 Energy Integral Equation 91

4.6.1 Enthalpy Thickness 93

4.6.2 Conduction Thickness 93

4.6.3 Convection Conductance or Heat Transfer Coefficient 93

4.7 Alternative Form of the Energy Integral Equation 94

4.8 Energy Integral Equation for Two-Dimensional Flow 94

Problems 94

References 96

5 Dimensional Analysis 97

5.1 Introduction 97

5.2 Dimensional Analysis 101

5.2.1 Dimensional Homogeneity 102

5.2.2 Buckingham π Theorem 102

5.2.3 Determination of π Terms 103

5.3 Nondimensionalization of Basic Differential Equations 116

5.4 Discussion 125

5.5 Dimensionless Numbers 125

5.5.1 Reynolds Number 125

5.5.2 Peclet Number 126

5.5.3 Prandtl Number 126

5.5.4 Nusselt Number 126

5.5.5 Stanton Number 126

5.5.6 Skin Friction Coefficient 126

5.5.7 Graetz Number 127

5.5.8 Eckert Number 127

5.5.9 Grashof Number 127

5.5.10 Rayleigh Number 127

5.5.11 Brinkman Number 127

5.6 Correlations of Experimental Data 128

Problems 136

References 147

6 One-Dimensional Solutions in Convective Heat Transfer 149

6.1 Introduction 149

6.2 Couette Flow 151

6.3 Poiseuille Flow 156

6.4 Rotating Flows 171

Problems 175

References 180

7 Laminar External Boundary Layers: Momentum and Heat Transfer 183

7.1 Introduction 183

7.2 Velocity Boundary Layer over a Semi-Infinite Flat Plate: Similarity Solution 183

7.2.0.1 x-Component of Velocity – u/ U∞ 190

7.2.0.2 Boundary Layer Thickness δ(x) 190

7.2.0.3 Wall Shear Stress τw 191

7.2.0.4 Local Skin Friction Coefficient cf (x) 191

7.2.0.5 drag Force d 192

7.2.0.6 Average Skin Friction Coefficient c̅f 192

7.2.0.7 Displacement Thickness δ1(x) 192

7.2.0.8 Momentum Thickness δ2(x) 192

7.3 Momentum Transfer over a Wedge (Falkner–Skan Wedge Flow): Similarity Solution 195

7.4 Application of Integral Methods to Momentum Transfer Problems 201

7.4.1 Laminar Forced Flow over a Flat Plate with Uniform Velocity 203

7.4.2 Two-Dimensional Laminar Flow over a Surface with Pressure Gradient (Variable Free Stream Velocity) 204

7.4.2.1 The Correlation Method of Thwaites 207

7.4.2.2 A Thwaites Type Correlation for Axisymmetric Body 212

7.5 Viscous Incompressible Constant Property Parallel Flow over a Semi-Infinite Flat Plate: Similarity Solution for Uniform Wall Temperature Boundary Condition 212

7.6 Low-Prandtl-Number Viscous Incompressible Constant Property Parallel Flow over a Semi-Infinite Flat Plate: Similarity Solutions for Uniform Wall Temperature Boundary Condition 225

7.7 High-Prandtl-Number Viscous Incompressible Constant Property Parallel Flow over a Semi-Infinite Flat Plate: Similarity Solutions for Uniform Wall Temperature Boundary Condition 228

7.8 Viscous Incompressible Constant Property Parallel Flow over a Semi-Infinite Flat Plate: Similarity Solution for Uniform Heat Flux Boundary Condition 230

7.9 Viscous Incompressible Constant Property Parallel Flow over a Semi-Infinite Flat Plate: Similarity Solutions for Variable Wall Temperature Boundary Condition 237

7.9.1 Superposition Principle 245

7.10 Viscous Incompressible Constant Property Flow over a Wedge (Falkner–Skan Wedge Flow): Similarity Solution for Uniform Wall Temperature Boundary Condition 249

7.11 Effect of Property Variation 252

7.12 Application of Integral Methods to Heat Transfer Problems 253

7.12.1 Viscous Flow with Constant Free Stream Velocity Along a Semi-Infinite Plate Under Uniform Wall Temperature: With Unheated Starting Length or Adiabatic Segment 256

7.12.1.1 The Plate Without Unheated Starting Length 262

7.12.2 Viscous Flow with Constant Free Stream Velocity Along a Semi-Infinite Plate with Uniform Wall Heat Flux: With Unheated Starting Length (Adiabatic Segment) 262

7.12.2.1 The Plate with No Unheated Starting Length 265

7.13 Superposition Principle 265

7.13.1 Superposition Principle Applied to Slug Flow over a Flat Plate: Arbitrary Variation in Wall Temperature 266

7.13.1.1 Boundary Condition: Single Step at X = 0 266

7.13.1.2 Boundary Condition: Two Steps at X = 0 and X =ξ1 268

7.13.1.3 Boundary Condition: Three Steps at X = 0, X =ξ1 , and X =ξ2 268

7.13.2 Superposition Principle Applied to Slug Flow over a Flat Plate: Arbitrary Variation in Wall Heat Flux 272

7.13.2.1 Boundary Condition: Single Step at X = 0 273

7.13.2.2 Boundary Condition: Two Steps at X = 0 and X =ξ1 274

7.13.2.3 Boundary Condition: Triple Steps at X = 0, X =ξ1 , and X =ξ2 275

7.13.3 Superposition Principle Applied to Viscous Flow over a Flat Plate: Stepwise Variation in Wall Temperature 278

7.13.3.1 First Problem 278

7.13.3.2 Second Problem 279

7.13.3.3 Heat Flux for 0 < X < ξ 279

7.13.3.4 The Heat Flux for X > ξ 1 280

7.13.4 Superposition Principle Applied to Viscus Flow over a Flat Plate: Stepwise Variation in Surface Heat Flux 282

7.13.4.1 First Problem 282

7.13.4.2 Second Problem 283

7.14 Viscous Flow over a Flat Plate with Arbitrary Surface Temperature Distribution 284

7.15 Viscous Flow over a Flat Plate with Arbitrarily Specified Heat Flux 289

7.16 One-Parameter Integral Method for Incompressible Two-Dimensional Laminar Flow Heat Transfer: Variable U ∞ (x) and Constant Tw − T∞ = const 293

7.17 One-Parameter Integral Method for Incompressible Laminar Flow Heat Transfer over a Constant Temperature of a Body of Revolution 295

Problems 299

References 310

8 Laminar Momentum and Heat Transfer in Channels 313

8.1 Introduction 313

8.2 Momentum Transfer 313

8.2.1 Hydrodynamic Considerations in Ducts 313

8.2.2 Fully Developed Laminar Flow in Circular Tube 318

8.2.3 Fully Developed Flow Between Two Infinite Parallel Plates 323

8.3 Thermal Considerations in Ducts 326

8.4 Heat Transfer in the Entrance Region of Ducts 335

8.4.1 Circular Pipe: Slug Flow Heat Transfer in the Entrance Region 337

8.4.1.1 Heat Transfer for Low-Prandtl-Number Fluid Flow (Slug Flow) in the Entrance Region of Circular Tube Subjected to Constant Wall Temperature 337

8.4.1.2 Heat Transfer to Low-Prandtl-Number Fluid Flow (Slug Flow) in the Entrance Region of the Circular Tube Subjected to Constant Heat Flux 345

8.4.1.3 Empirical and Theoretical Correlations for Viscous Flow Heat Transfer in the Entrance Region of the Circular Tube 350

8.4.2 Parallel Plates: Slug Flow Heat Transfer in the Entrance Region 355

8.4.2.1 Heat Transfer to a Low-Prandtl-Number Fluid (Slug Flow) in the Entrance Region of Parallel Plates: Both Plates Are Subjected to Constant Wall Temperatures 355

8.4.2.2 Heat Transfer for Low-Prandtl-Number Fluid Flow (Slug Flow) in the Entrance Region of Parallel Plates: Both Plates Are Subjected to UHF 358

8.4.2.3 Heat Transfer for Low-Prandtl-Number Fluid Flow (Slug Flow) in the Entrance Region of Parallel Plates: Upper Plate Is Insulated While the Lower Plate Is Subjected to Constant Wall Temperature 363

8.4.2.4 Heat Transfer for Low-Prandtl-Number Fluid Flow (Slug Flow) in the Entrance Region of Parallel Plates: Upper Plate Is Insulated While the Lower Plate Is Subjected to Constant Heat Flux 367

8.4.2.5 Empirical and Theoretical Correlations for Viscous Flow Heat Transfer in the Entrance Region of Parallel Plates 370

8.5 Fully Developed Heat Transfer 372

8.5.1 Circular Tube 372

8.5.1.1 HFD and TFD Laminar Forced Convection Heat Transfer for Slug Flow in a Circular Pipe Subjected to Constant Wall Heat Flux 372

8.5.1.2 HFD and TFD Laminar Forced Convection Heat Transfer for Viscous Flow in a Circular Tube Subjected to Constant Wall Heat Flux 375

8.5.1.3 HFD and TFD Laminar Forced Convection Heat Transfer for Viscous Flow in a Circular Tube Subjected to Constant Wall Temperature 378

8.5.2 Infinite Parallel Plates 382

8.5.2.1 HFD and TFD Laminar Forced Convection Heat Transfer for Viscous Flow Between a Parallel Plate Channel. Both Plates Are Subjected to Constant Wall Heat Flux Boundary Condition 383

8.6 Heat Transfer in the Thermal Entrance Region 387

8.6.1 Circular Tube 388

8.6.1.1 Graetz Problem: HFD and Thermally Developing Flow in a Circular Tube under Constant Wall Temperature Boundary Condition 388

8.6.1.2 The Leveque Solution: UWT Boundary Condition 401

8.6.1.3 Graetz Problem: HFD and Thermally Developing Flow for Viscous Flow in Circular Tube Under Uniform Wall Heat Flux Boundary Condition 406

8.6.1.4 Empirical and Theoretical Correlations for Viscous Flow in the Thermal Entrance Region of the Pipe 415

8.6.2 Two Infinite Parallel Plates 419

8.6.2.1 Graetz Problem: HFD and Thermally Developing Flow Between Parallel Plates Subjected to Constant Wall Temperature 419

8.6.2.2 Graetz Problem: HFD and Thermally Developing Flow Between Parallel Plates Subjected to Constant Wall Heat Flux 428

8.6.2.3 Empirical and Theoretical Correlations for Viscous Flow in Thermal Entrance Region of Parallel Plates 436

8.7 Circular Pipe with Variable Surface Temperature Distribution in the Axial Direction 438

8.8 Circular Pipe with Variable Surface Heat Flux Distribution in the Axial Direction 443

8.9 Short Tubes 446

8.10 Effect of Property Variation 448

8.11 Regular Sturm-Liouville Systems 449

Problems 450

References 463

9 Foundations of Turbulent Flow 465

9.1 Introduction 465

9.2 The Reynolds Experiment 465

9.3 Nature of Turbulence 466

9.4 Time Averaging and Fluctuations 467

9.5 Isotropic Homogeneous Turbulence 470

9.6 Reynolds Averaging 470

9.7 Governing Equations of Incompressible Steady Mean Turbulent Flow 474

9.8 Turbulent Momentum Boundary Layer Equation 477

9.9 Turbulent Energy Equation 478

9.10 Turbulent Boundary Layer Energy Equation 479

9.11 Closure Problem of Turbulence 480

9.12 Eddy Diffusivity of Momentum 481

9.13 Eddy Diffusivity of Heat 482

9.14 Transport Equations in the Cylindrical Coordinate System 483

9.15 Experimental Work on the Turbulent Mean Flow 484

9.15.1 Turbulent Flow in Pipe: Velocity Profiles 485

9.15.2 Turbulent Flow over a Flat Plate: Velocity Profiles 491

9.16 Transition to Turbulent Flow 496

Problems 498

References 504

10 Turbulent External Boundary Layers: Momentum and Heat Transfer 507

10.1 Introduction 507

10.2 Turbulent Momentum Boundary Layer 507

10.3 Turbulence Models 508

10.3.1 Zero-Equation Models 508

10.3.1.1 Boussinesq Model 508

10.3.1.2 Prandtl’s Mixing-Length Model 508

10.3.1.3 Van Driest Model 509

10.4 Turbulent Flow over a Flat Plate with Constant Free-Stream Velocity: Couette Flow Approximation 510

10.4.1 Inner Region 510

10.5 The Universal Velocity Profile 511

10.5.1 Three-Layer (von Karman) Model for the Velocity Profile 511

10.5.2 Other Velocity Models 514

10.6 Approximate Solution by the Integral Method for the Turbulent Momentum Boundary Layer over a Flat Plate 514

10.7 Laminar and Turbulent Boundary Layer 519

10.8 Other Eddy Diffusivity Momentum Models 521

10.9 Turbulent Heat Transfer 522

10.10 Analogy Between Momentum and Heat Transfer 529

10.10.1 Reynold’s Analogy 529

10.10.2 Chilton–Colburn Analogy 531

10.10.3 Prandtl–Taylor Analogy 532

10.10.4 Von Karman Analogy 535

10.11 Some Other Correlations for Turbulent Flow over a Flat Plate 539

10.12 Turbulent Flow Along a Semi-infinite Plate with Unheated Starting Length: Constant Temperature Solution 542

10.13 Flat Plate with Arbitrarily Specified Surface Temperature 550

10.14 Constant Free-Stream Velocity Flow Along a Flat Plate with Uniform Heat Flux 553

10.15 Turbulent Flow Along a Semi-Infinite Plate with Arbitrary Heat Flux Distribution 554

10.16 Turbulent Transition and Overall Heat Transfer 558

10.17 Property Variation 564

Problems 564

References 569

11 Turbulent Internal Flow: Momentum and Heat Transfer 573

11.1 Introduction 573

11.2 Momentum Transfer 573

11.2.1 Momentum Transfer in Infinite Two Parallel Plates 573

11.2.1.1 The Entrance Region 574

11.2.1.2 The HFD Region 575

11.2.1.3 Prandtl’s Mixing-Length Model 578

11.2.1.4 Buffer Region 579

11.2.1.5 The Mean Velocity 582

11.2.1.6 Skin Friction Coefficient or Fanning Friction Factor cf 582

11.2.2 Momentum Transfer in Circular Pipe Flow 585

11.2.2.1 Entrance Region 585

11.2.2.2 The HFD Region 586

11.2.2.3 Average Velocity V 589

11.2.2.4 Skin Friction Factor cf 589

11.2.2.5 Moody Friction Factor f 589

11.2.2.6 Prandtl Mixing-Length Model 590

11.2.2.7 Laminar Sublayer 591

11.2.2.8 Buffer Region 591

11.2.2.9 Turbulent Region 591

11.2.2.10 Moody Friction Factor 592

11.2.2.11 Fanning Friction Factor 593

11.2.2.12 The Power Law Velocity Distribution 596

11.3 Fully Developed Turbulent Heat Transfer 597

11.3.1 TFD and HFD Turbulent Flow Between Parallel Plates Subjected to UHF 598

11.3.1.1 Mean Stream Temperature 602

11.3.2 TFD and HFD Turbulent Flow in a Pipe Subjected to UHF 605

11.3.2.1 Laminar Viscous Sublayer: 0 < y+ < 5 609

11.3.2.2 Buffer Layer: 5 < y+ < 30 610

11.3.2.3 Turbulent Region: y+ > 30 610

11.4 HFD Thermally Developing Turbulent Heat Transfer 618

11.4.1 Circular Duct with UWT 618

11.4.2 Circular Duct with Uniform Wall Heat Flux 625

11.4.2.1 Solution for Fully Developed Temperature Distribution θ1 626

11.4.2.2 Solution for the Entry Region Temperature Distribution θ2 627

11.5 Analogies for Internal Flow 629

11.5.1 Reynolds Analogy 629

11.5.2 Colburn Analogy 631

11.5.3 Prandtl–Taylor Analogy 631

11.5.3.1 Laminar Sublayer 632

11.5.3.2 Turbulent Core 632

11.5.4 von Karman Analogy 633

11.5.4.1 Laminar Sublayer: 0 ≤ y+ ≪ 5 634

11.5.4.2 Buffer Layer: 5 ≤ y+ ≪ 30 635

11.5.4.3 Turbulent Core: y+ ≥ 30 635

11.5.5 The Analogy of Kadar and Yaglom 636

11.5.6 The Analogy of Yu et al. 637

11.5.7 Martinelli Analogy 639

11.6 Combined Entrance Region 641

11.7 Empirical and Theoretical Correlations for Turbulent Flow in Channels 642

11.7.1.1 Colburn Correlation 645

11.7.1.2 Dittus and Boelter Correlation 646

11.7.1.3 Sieder–Tate Correlation 646

11.7.1.4 Hausen Correlations 647

11.7.1.5 Petukhov Correlation 647

11.7.1.6 Gnielinski Correlation 649

11.7.1.7 Gnielinski Correlation with Modification 650

11.7.1.8 Sleicher and Rouse Correlation 650

11.7.1.9 Nusselt Correlation 651

11.8 Heat Transfer in Transitional Flow 652

11.8.1 Friction Factor in the Transitional Flow 653

11.8.2 Heat Transfer in the Transition Region 654

11.8.2.1 Tam and Ghajar Approach 654

11.8.2.2 Churchill Approach 655

11.8.2.3 Gnielinski Approach 656

11.8.2.4 Abraham et al. Approach 657

11.9 Effect of Property Variation 660

Problems 660

References 670

12 Free Convection Heat Transfer 675

12.1 Introduction 675

12.2 Fundamental Equations and Dimensionless Parameters of Free Convection 675

12.3 Scaling in Natural Convection 679

12.4 Similarity Solution for Laminar Boundary Layer over a Semi-Infinite Vertical Flat Plate 681

12.4.1 Constant Wall Temperature 681

12.4.2 Uniform Heat Flux 688

12.5 Integral Method (von Karman–Pohlhausen Method): An Approximate Analysis of Laminar Free Convection on a Vertical Plate 695

12.5.1 Constant Wall Temperature 697

12.5.2 Uniform Heat Flux 700

12.6 Turbulent Free Convection Heat Transfer on a Vertical Plate 702

12.7 Empirical Correlations for Free Convection 704

12.7.1 Vertical Plate 704

12.7.2 Horizontal Plate 712

12.7.3 Inclined Plates 715

12.7.4 Vertical Cylinders 719

12.7.5 Horizontal Cylinder 722

12.7.6 Inclined Cylinder 723

12.7.7 Free Convection from Vertical Cylinders of Small Diameter 724

12.8 Free Convection Within Parallel Plate Channels 725

12.8.1 Vertical Parallel Plate Channel 725

12.8.2 Horizontal Parallel Plate Channel 731

12.8.3 Inclined Parallel Plate Channel 732

12.9 Rectangular Enclosures 735

12.9.1 Horizontal Rectangular Enclosure (θ=0) 735

12.9.2 Vertical Rectangular Enclosure 737

12.9.3 Inclined Rectangular Enclosure 740

12.10 Horizontal Concentric Cylinders 743

12.11 Concentric Spheres 744

12.12 Spheres 744

Problems 745

References 752

Index 755