An integrated, modern approach to transport phenomena for graduate students, featuring traditional and contemporary examples to demonstrate the diverse practical applications of the theory. Written in an easy to follow style, the basic principles of transport phenomena, and model building are recapped in Chapters 1 and 2 before progressing logically through more advanced topics including physicochemical principles behind transport models. Treatments of numerical, analytical, and computational solutions are presented side by side, often with sample code in MATLAB, to aid students' understanding and develop their confidence in using computational skills to solve real-world problems. Learning objectives and mathematical prerequisites at the beginning of chapters orient students to what is required in the chapter, and summaries and over 400 end-of-chapter problems help them retain the key points and check their understanding. Online supplementary material including solutions to problems for instructors, supplementary reading material, sample computer codes, and case studies complete the package.

Preface page xvii

Topical outline xxi

Notation xxiii

1 Introduction 1

1.1 What, why, and how? 2

      1.1.1 What? 2

      1.1.2 Why? 3

      1.1.3 How? 6

      1.1.4 Conservation statement 6

      1.1.5 The need for constitutive models 7

      1.1.6 Common constitutive models 8

1.2 Typical transport property values 10

      1.2.1 Viscosity: pure gases and vapors 10

      1.2.2 Viscosity: liquids 11

      1.2.3 Thermal conductivity 11

      1.2.4 Diffusivity 12

1.3 The continuum assumption and the field variables 13

      1.3.1 Continuum and pointwise representation 13

      1.3.2 Continuum vs. molecular 16

      1.3.3 Primary field variables 16

      1.3.4 Auxiliary variables 16

1.4 Coordinate systems and representation of vectors 18

      1.4.1 Cartesian coordinates 18

      1.4.2 Cylindrical coordinates 19

      1.4.3 Spherical coordinates 20

      1.4.4 Gradient of a scalar field 20

1.5 Modeling at various levels 22

      1.5.1 Levels based on control-volume size 22

      1.5.2 Multiscale models 24

      1.5.3 Multiscale modeling below the continuum level 25

1.6 Model building: general guidelines 25

1.7 An example application: pipe flow and tubular reactor 27

      1.7.1 Pipe flow: momentum transport 28

      1.7.2 Laminar or turbulent? 28

      1.7.3 Use of dimensionless numbers 30

      1.7.4 Pipe flow: heat transport 32

      1.7.5 Pipe flow: mass exchanger 35

      1.7.6 Pipe flow: chemical reactor 35

1.8 The link between transport properties and molecular models 36

      1.8.1 Kinetic theory concepts 37

      1.8.2 Liquids 42

      1.8.3 Transport properties of solids 44

1.9 Six decades of transport phenomena 45

1.10 Closure 48

Summary 49

Additional Reading 50

Problems 50

2 Examples of transport and system models 56

2.1 Macroscopic mass balance 58

      2.1.1 Species balance equation 58

      2.1.2 Transient balance: tracer studies 63

      2.1.3 Overall mass balance 65

2.2 Compartmental models 68

      2.2.1 Model equations 68

      2.2.2 Matrix representation 69

      2.2.3 A numerical IVP solver in MATLAB 70

2.3 Macroscopic momentum balance 72

      2.3.1 Linear momentum 72

      2.3.2 Angular momentum 77

2.4 Macroscopic energy balances 79

      2.4.1 Single inlet and outlet 79

      2.4.2 The Bernoulli equation 81

      2.4.3 Sonic and subsonic flows 85

      2.4.4 Cooling of a solid: a lumped model 91

2.5 Examples of differential balances: Cartesian 97

      2.5.1 Heat transfer with nuclear fission in a slab 97

      2.5.2 Mass transfer with reaction in a porous catalyst 99

      2.5.3 Momentum transfer: unidirectional flow in a channel 101

2.6 Examples of differential models: cylindrical coordinates 102

      2.6.1 Heat transfer with generation 102

      2.6.2 Mass transfer with reaction 104

      2.6.3 Flow in a pipe 105

2.7 Spherical coordinates 106

2.8 Examples of mesoscopic models 108

      2.8.1 Tubular reactor with heat transfer 108

      2.8.2 Heat transfer in a pin fin 109

      2.8.3 Countercurrent heat exchanger 110

      2.8.4 Counterflow: matrix method 115

Summary 116

Problems 119

3 Flow kinematics 126

3.1 Eulerian description of velocity 128

3.2 Lagrangian description: the fluid particle 128

3.3 Acceleration of a fluid particle 130

3.4 The substantial derivative 130

3.5 Dilatation of a fluid particle 132

3.6 Mass continuity 134

3.7 The Reynolds transport theorem 135

3.8 Vorticity and rotation 136

3.8.1 Curl in other coordinate systems 137

3.8.2 Circulation along a closed curve 139

3.9 Vector potential representation 140

3.10 Streamfunctions 141

      3.10.1 Two-dimensional flows: Cartesian 141

      3.10.2 Two-dimensional flows: polar 143

      3.10.3 Streamfunctions in axisymmetric flows 143

      3.10.4 The relation to vorticity: the E2 operator 144

3.11 The gradient of velocity 145

      3.12 Deformation and rate of strain 146

      3.12.1 The physical meaning of the rate of strain 148

      3.12.2 Rate of strain: cylindrical 151

      3.12.3 Rate of strain: spherical 151

      3.12.4 Invariants of a tensor 152

      3.13 Index notation for vectors and tensors 152

Summary 154

Problems 155

4 Forces and their representations 159

4.1 Forces on fluids and their representation 160

      4.1.1 Pressure forces 161

      4.1.2 Viscous forces 163

      4.1.3 The divergence of a tensor 167

4.2 The equation of hydrostatics 169

      4.2.1 Archimedes’ principle 169

      4.2.2 The force on a submerged surface: no curvature 170

      4.2.3 Force on a curved surface 171

4.3 Hydrostatics at interfaces 172

      4.3.1 The nature of interfacial forces 172

      4.3.2 Contact angle and capillarity 174

      4.3.3 The Laplace–Young equation 175

4.4 Drag and lift forces 177

Summary 180

Problems 181

5 Equations of motion and the Navier–Stokes equation 184

5.1 Equation of motion: the stress form 185

      5.1.1 The Lagrangian point particle 185

      5.1.2 The Lagrangian control volume 186

      5.1.3 The Eulerian control volume 187

5.2 Types of fluid behavior 189

      5.2.1 Types and classification of fluid behavior 189

      5.2.2 Stress relations for a Newtonian fluid 191

5.3 The Navier–Stokes equation 191

      5.3.1 The Laplacian of velocity 192

      5.3.2 Common boundary conditions for flow problems 193

5.4 The dimensionless form of the flow equation 195

      5.4.1 Key dimensionless groups 195

      5.4.2 The Stokes equation: slow flow or viscous flow 196

      5.4.3 The Euler equation 197

5.5 Use of similarity for scaleup 197

5.6 Alternative representations for the Navier–Stokes equations 201

      5.6.1 Plane flow: the vorticity–streamfunction form 201

      5.6.2 Plane flow: the streamfunction representation 201

      5.6.3 Inviscid and potential flow 202

      5.6.4 The velocity–vorticity formulation 202

      5.6.5 Slow flow in terms of vorticity 202

      5.6.6 The pressure Poisson equation 203

5.7 Constitutive models for non-Newtonian fluids 203

Summary 205

Problems 206

6 Illustrative flow problems 208

6.1 Introduction 210

      6.1.1 Summary of equations 210

      6.1.2 Simplifications 211

      6.1.3 Solution methods 211

6.2 Channel flow 212

      6.2.1 Entry-region flow in channels or pipes 212

      6.2.2 General solution 214

      6.2.3 Pressure-driven flow 215

      6.2.4 Shear-driven flow 215

      6.2.5 Gravity-driven flow 216

6.3 Axial flow in cylindrical geometry 218

      6.3.1 Circular pipe 219

      6.3.2 Annular pipe: pressure-driven 219

      6.3.3 Annular pipe: shear-driven 220

6.4 Torsional flow 220

6.5 Radial flow 222

6.6 Flow in a spherical gap 223

6.7 Non-circular channels 224

6.8 The lubrication approximation 227

      6.8.1 Flow between two inclined plates 227

      6.8.2 Flow in a tapered pipe 228

6.9 External flow 230

      6.10 Non-Newtonian viscoinelastic fluids 233

      6.10.1 A power-law model 233

      6.10.2 Flow of a Bingham fluid in a pipe 234

      6.10.3 The Rabinowitsch equation 236

6.11 The effect of fluid elasticity 237

6.12 A simple magnetohydrodynamic problem 240

Summary 244

Additional Reading 246

Problems 246

7 The energy balance equation 251

7.1 Application of the first law of thermodynamics to a moving control volume 252

7.2 The working rate of the forces 253

7.3 Kinetic energy and internal energy equations 256

7.4 The enthalpy form 257

7.5 The temperature equation 257

7.6 Common boundary conditions 259

7.7 The dimensionless form of the heat equation 261

7.8 From differential to macroscopic 262

7.9 Entropy balance and the second law of thermodynamics 263

      7.9.1 Some definitions from thermodynamics 263

Summary 267

Problems 268

8 Illustrative heat transport problems 269

8.1 Steady heat conduction and no generation 270

      8.1.1 Constant conductivity 270

      8.1.2 Variable thermal conductivity 273

      8.1.3 Two-dimensional heat conduction problems 274

8.2 Heat conduction with generation: the Poisson equation 276

      8.2.1 The constant-generation case 276

8.3 Conduction with temperature-dependent generation 277

      8.3.1 Linear variation with temperature 277

      8.3.2 Non-linear variation with temperature 279

      8.3.3 Two-dimensional Poisson problems 281

8.4 Convection effects 282

      8.4.1 Transpiration cooling 282

      8.4.2 Convection in boundary layers 285

8.5 Mesoscopic models 286

      8.5.1 Heat transfer from a fin 286

      8.5.2 A single-stream heat exchanger 288

8.6 Volume averaging or lumping 290

      8.6.1 Cooling of a sphere in a liquid 290

      8.6.2 An improved lumped model 291

Summary 292

Problems 293

9 Equations of mass transfer 296

9.1 Preliminaries 298

9.2 Concentration jumps at interfaces 300

9.3 The frame of reference and Fick’s law 302

9.4 Equations of mass transfer 307

      9.4.1 Mass basis 308

      9.4.2 Mole basis 310

      9.4.3 Boundary conditions 311

9.5 From differential to macroscopic 312

9.6 Complexities in diffusion 313

Summary 316

Problems 317

10 Illustrative mass transfer problems 321

10.1 Steady-state diffusion: no reaction 322

      10.1.1 Summary of equations 322

10.2 The film concept in mass-transfer analysis 328

      10.2.1 Fluid–solid interfaces 328

      10.2.2 Gas–liquid interfaces: the two-film model 331

10.3 Mass transfer with surface reaction 333

      10.3.1 Heterogeneous reactions: the film model 333

10.4 Mass transfer with homogeneous reactions 334

      10.4.1 Diffusion in porous media 334

      10.4.2 Diffusion and reaction in a porous catalyst 335

      10.4.3 First-order reaction 335

      10.4.4 Zeroth-order reaction 339

      10.4.5 Transport in tissues: the Krogh model 340

      10.4.6 mth-order reaction 342

10.5 Models for gas–liquid reaction 343

      10.5.1 Analysis for the pseudo-first-order case 346

      10.5.2 Analysis for instantaneous asymptote 347

      10.5.3 The second-order case: an approximate solution 347

      10.5.4 The instantaneous case: the effect of gas film resistance 348

10.6 Transport across membranes 350

      10.6.1 Gas transport: permeability 350

      10.6.2 Complexities in membrane transport 352

      10.6.3 Liquid-separation membranes 353

10.7 Transport in semi-permeable membranes 354

      10.7.1 Reverse osmosis 355

      10.7.2 Concentration-polarization effects 356

      10.7.3 The Kedem–Katchalsky model 358

      10.7.4 Transport in biological membranes 360

10.8 Reactive membranes and facilitated transport 360

      10.8.1 Reactive membrane: facilitated transport 360

      10.8.2 Co- and counter-transport 363

10.9 A boundary-value solver in MATLAB 364

      10.9.1 Code-usage procedure 364

      10.9.2 BVP4C example: the selectivity of a catalyst 364

Summary 367

Additional Reading 370

Problems 370

11 Analysis and solution of transient transport processes 377

11.1 Transient conduction problems in one dimension 378

11.2 Separation of variables: the slab with Dirichlet conditions 380

      11.2.1 Slab: temperature profiles 383

      11.2.2 Slab: heat flux 384

      11.2.3 Average temperature 384

11.3 Solutions for Robin conditions: slab geometry 385

11.4 Robin case: solutions for cylinder and sphere 387

11.5 Two-dimensional problems: method of product solution 388

11.6 Transient non-homogeneous problems 389

      11.6.1 Subtracting the steady-state solution 390

      11.6.2 Use of asymptotic solution 391

11.7 Semi-infinite-slab analysis 391

      11.7.1 Constant surface temperature 392

      11.7.2 Constant flux and other boundary conditions 393

11.8 The integral method of solution 394

11.9 Transient mass diffusion 396

      11.9.1 Constant diffusivity model 396

      11.9.2 The penetration theory of mass transfer 399

      11.9.3 The effect of chemical reaction 399

      11.9.4 Variable diffusivity 403

11.10 Periodic processes 404

      11.10.1 Analysis for a semi-infinite slab 405

      11.10.2 Analysis for a finite slab 407

11.11 Transient flow problems 408

      11.11.1 Start-up of channel flow 409

      11.11.2 Transient flow in a semi-infinite mass of fluid 409

      11.11.3 Flow caused by an oscillating plate 409

      11.11.4 Start-up of Poiseuille flow 411

      11.11.5 Pulsatile flow in a pipe 412

11.12 A PDE solver in MATLAB 413

      11.12.1 Code usage 413

      11.12.2 Example general code for 1D transient conduction 415

Summary 417

Additional Reading 418

Problems 419

12 Convective heat and mass transfer 425

12.1 Heat transfer in laminar flow 427

      12.1.1 Preliminaries and the model equations 427

      12.1.2 The constant-wall-temperature case: the Graetz problem 430

      12.1.3 The constant-flux case 434

12.2 Entry-region analysis 435

      12.2.1 The constant-wall-temperature case 435

      12.2.2 The constant-flux case 437

12.3 Mass transfer in film flow 437

      12.3.1 Solid dissolution at a wall in film flow 438

      12.3.2 Gas absorption from interfaces in film flow 439

12.4 Laminar-flow reactors 440

      12.4.1 A 2D model and key dimensionless groups 440

      12.4.2 The pure convection model 443

12.5 Laminar-flow reactor: a mesoscopic model 444

      12.5.1 Averaging and the concept of dispersion 444

      12.5.2 Non-linear reactions 446

12.6 Numerical study examples with PDEPE 446

      12.6.1 The Graetz problem 446

Summary 449

Problems 450

13 Coupled transport problems 453

13.1 Modes of coupling 454

      13.1.1 One-way coupling 454

      13.1.2 Two-way coupling 455

13.2 Natural convection problems 455

      13.2.1 Natural convection between two vertical plates 455

      13.2.2 Natural convection over a vertical plate 459

      13.2.3 Natural convection: concentration effects 460

13.3 Heat transfer due to viscous dissipation 460

      13.3.1 Viscous dissipation in plane Couette flow 460

      13.3.2 Laminar heat transfer with dissipation: the Brinkman problem 461

13.4 Laminar heat transfer: the effect of viscosity variations 463

13.5 Simultaneous heat and mass transfer: evaporation 465

      13.5.1 Dry- and wet-bulb temperatures 465

      13.5.2 Evaporative or sweat cooling 468

13.6 Simultaneous heat and mass transfer: condensation 468

      13.6.1 Condensation of a vapor in the presence of a non-condensible gas 468

      13.6.2 Fog formation 472

      13.6.3 Condensation of a binary gas mixture 472

13.7 Temperature effects in a porous catalyst 476

Summary 480

Additional Reading 481

Problems 481

14 Scaling and perturbation analysis 484

14.1 Dimensionless analysis revisited 485

      14.1.1 The method of matrix transformation 486

      14.1.2 Momentum problems 486

      14.1.3 Energy transfer problems 489

      14.1.4 Mass transfer problems 491

      14.1.5 Example: scaleup of agitated vessels 492

      14.1.6 Example: pump performance correlation 493

14.2 Scaling analysis 495

      14.2.1 Transient diffusion in a semi-infinite region 495

      14.2.2 Example: gas absorption with reaction 496

      14.2.3 Kolmogorov scales for turbulence: an example of scaling 496

      14.2.4 Scaling analysis of flow in a boundary layer 497

      14.2.5 Flow over a rotating disk 501

14.3 Perturbation methods 503

      14.3.1 Regular perturbation 503

      14.3.2 The singular perturbation method 506

      14.3.3 Example: catalyst with spatially varying activity 507

      14.3.4 Example: gas absorption with reversible reaction 508

      14.3.5 Stokes flow past a sphere: the Whitehead paradox 511

14.4 Domain perturbation methods 513

Summary 515

Additional Reading 516

Problems 516

15 More flow analysis 523

15.1 Low-Reynolds-number (Stokes) flows 525

      15.1.1 Properties of Stokes flow 525

15.2 The mathematics of Stokes flow 527

      15.2.1 General solutions: spherical coordinates 527

      15.2.2 Flow past a sphere: use of the general solution 528

      15.2.3 Bubbles and drops 531

      15.2.4 Oseen’s improvement 533

      15.2.5 Viscosity of suspensions 534

      15.2.6 Nanoparticles: molecular effects 535

15.3 Inviscid and irrotational flow 536

      15.3.1 Properties of irrotational flow 536

      15.3.2 The Bernoulli equation revisited 537

15.4 Numerics of irrotational flow 539

      15.4.1 Boundary conditions 539

      15.4.2 Solutions using harmonic functions 540

      15.4.3 Solution using singularities 542

15.5 Flow in boundary layers 546

      15.5.1 Relation to the vorticity transport equation 547

      15.5.2 Flat plate: integral balance 548

      15.5.3 The integral method: the von Kármán method 549

      15.5.4 The average value of drag 550

      15.5.5 Non-flat systems: the effect of a pressure gradient 550

15.6 Use of similarity variables 551

      15.6.1 A simple computational scheme 553

      15.6.2 Wedge flow: the Falkner–Skan equation 554

      15.6.3 Blasius flow 554

      15.6.4 Stagnation-point (Hiemenz) flow 555

15.7 Flow over a rotating disk 556

Summary: Stokes flow 557

Summary: potential flow 558

Summary: boundary-layer theory 558

Additional Reading 559

Problems 559

16 Bifurcation and stability analysis 566

16.1 Introduction to dynamical systems 567

      16.1.1 Arc-length continuation: a single-equation example 571

      16.1.2 The arc-length method: multiple equations 572

16.2 Bifurcation and multiplicity of DPSs 576

      16.2.1 A bifurcation example: the Frank-Kamenetskii equation 576

      16.2.2 Bifurcation: porous catalyst 577

16.3 Flow-stability analysis 578

      16.3.1 Evolution equations and linearized form 578

      16.3.2 Normal-mode analysis 580

16.4 Stability of shear flows 581

      16.4.1 The Orr–Sommerfeld equation 581

      16.4.2 Stability of shear layers: the role of viscosity 583

      16.4.3 The Rayleigh equation 583

      16.4.4 Computational methods 584

16.5 More examples of flow instability 585

      16.5.1 Kelvin–Helmholtz instability 585

      16.5.2 Rayleigh–Taylor instability 586

      16.5.3 Thermal instability: the Bénard problem 587

      16.5.4 Marangoni instability 588

      16.5.5 Non-Newtonian fluids 588

Summary 589

Additional Reading 589

Problems 589

17 Turbulent-flow analysis 592

17.1 Flow transition and properties of turbulent flow 593

17.2 Time averaging 594

17.3 Turbulent heat and mass transfer 597

17.4 Closure models 598

17.5 Flow between two parallel plates 599

17.6 Pipe flow 603

      17.6.1 The effect of roughness 605

17.7 Turbulent boundary layers 606

17.8 Other closure models 607

      17.8.1 The two-equation model: the k− model 608

      17.8.2 Reynolds-stress models 609

      17.8.3 Large-eddy simulation 610

      17.8.4 Direct numerical simulation 610

17.9 Isotropy, correlation functions, and the energy spectrum 610

17.10 Kolmogorov’s energy cascade 613

      17.10.1 Correlation in the spectral scale 614

Summary 615

Additional Reading 616

Problems 616

18 More convective heat transfer 619

18.1 Heat transport in laminar boundary layers 620

      18.1.1 Problem statement and the differential equation 620

      18.1.2 The thermal boundary layer: scaling analysis 621

      18.1.3 The heat integral equation 624

      18.1.4 Thermal boundary layers: similarity solution 627

18.2 Turbulent heat transfer in channels and pipes 628

      18.2.1 Pipe flow: the Stanton number 633

18.3 Heat transfer in complex geometries 635

18.4 Natural convection on a vertical plate 636

      18.4.1 Natural convection: computations 640

18.5 Boiling systems 641

      18.5.1 Pool boiling 641

      18.5.2 Nucleate boiling 641

18.6 Condensation problems 645

18.7 Phase-change problems 647

Summary 650

Additional reading 651

Problems 651

19 Radiation heat transfer 656

19.1 Properties of radiation 657

19.2 Absorption, emission, and the black body 657

19.3 Interaction between black surfaces 661

19.4 Gray surfaces: radiosity 664

19.5 Calculations of heat loss from gray surfaces 666

19.6 Radiation in absorbing media 670

Summary 674

Additional Reading 675

Problems 675

20 More convective mass transfer 678

20.1 Mass transfer in laminar boundary layers 679

      20.1.1 The low-flux assumption 679

      20.1.2 Dimensional analysis 680

      20.1.3 Scaling analysis 681

      20.1.4 The low-flux case: integral analysis 682

      20.1.5 The low-flux case: exact analysis 685

20.2 Mass transfer: the high-flux case 686

      20.2.1 The film model revisited 686

      20.2.2 The high-flux case: the integral-balance model 688

      20.2.3 The high-flux case: the similarity-solution method 689

20.3 Mass transfer in turbulent boundary layers 689

20.4 Mass transfer at gas–liquid interfaces 691

      20.4.1 Turbulent films 691

      20.4.2 Single bubbles 692

      20.4.3 Bubble swarms 693

20.5 Taylor dispersion 693

Summary 696

Additional Reading 696

Problems 697

21 Mass transfer: multicomponent systems 700

21.1 A constitutive model for multicomponent transport 701

      21.1.1 Stefan–Maxwell models 701

      21.1.2 Generalization 702

21.2 Non-reacting systems and heterogeneous reactions 703

      21.2.1 Evaporation in a ternary mixture 703

      21.2.2 Evaporation of a binary liquid mixture 704

      21.2.3 Ternary systems with heterogeneous reactions 707

21.3 Application to homogeneous reactions 709

      21.3.1 Multicomponent diffusion in a porous catalyst 709

      21.3.2 MATLAB implementation 710

21.4 Diffusion-matrix-based methods 713

21.5 An example of pressure diffusion 717

21.6 An example of thermal diffusion 719

Summary 720

Additional Reading 721

Problems 721

22 Mass transport in charged systems 725

22.1 Transport of charged species: preliminaries 726

      22.1.1 Mobility and diffusivity 726

      22.1.2 The Nernst–Planck equation 727

      22.1.3 Potential field and charge neutrality 728

22.2 Electrolyte transport across uncharged membranes 732

22.3 Electrolyte transport in charged membranes 734

22.4 Transport effects in electrodialysis 735

22.5 Departure from electroneutrality 738

22.6 Electro-osmosis 741

22.7 The streaming potential 744

22.8 The sedimentation potential 746

22.9 Electrophoresis 747

22.10 Transport in ionized gases 748

Summary 750

Additional Reading 751

Problems 751

Closure 757

References 758

Index 766

P. A. Ramachandran is a Professor in the Department of Energy, Environment and Chemical Engineering at Washington University, St Louis. He has extensive teaching experience, mainly in transport phenomena, mathematical methods and chemical reaction engineering, and he has also held many visiting appointments at various international institutions. He has written or co-written two previous books, as well as over 200 journal articles in which he has pioneered many new concepts and computational tools for modelling of chemical reactors. He is the recipient of the Moulton Medal from the Institution of Chemical Engineers, UK, the NASA certificate of recognition, USA, and the NEERI award from the Institution of Chemical Engineers, India.

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