Scaling Analysis in Modeling Transport and Reaction Processes

This book is unique as the first effort to expound on the subject of systematic scaling analysis. Not written for a specific discipline, the book targets any reader interested in transport phenomena and reaction processes. The book is logically divided into chapters on the use of systematic scaling analysis in fluid dynamics, heat transfer, mass transfer, and reaction processes. An integrating chapter is included that considers more complex problems involving combined transport phenomena. Each chapter includes several problems that are explained in considerable detail. These are followed by several worked examples for which the general outline for the scaling is given. Each chapter also includes many practice problems.

This book is based on recognizing the value of systematic scaling analysis as a pedagogical method for teaching transport and reaction processes and as a research tool for developing and solving models and in designing experiments. Thus, the book can serve as both a textbook and a reference book.

Preface xi

Acknowledgments xv

1 Introduction 1

1.1 Motivation for Using Scaling Analysis 1

1.2 Organization of the Book 5

2 Systematic Method for Scaling Analysis 7

2.1 Introduction 7

2.2 Mathematical Basis for Scaling Analysis 7

2.3 Order-of-One Scaling Analysis 8

2.4 Scaling Alternative for Dimensional Analysis 13

2.5 Summary 18

3 Applications in Fluid Dynamics 19

3.1 Introduction 19

3.2 Fully Developed Laminar Flow 20

3.3 Creeping- and Lubrication-Flow Approximations 26

3.4 Boundary-Layer-Flow Approximation 32

3.5 Quasi-Steady-State-Flow Approximation 38

3.6 Flows with End and Sidewall Effects 43

3.7 Free Surface Flow 45

3.8 Porous Media Flow 52

3.9 Compressible Fluid Flow 56

3.10 Dimensional Analysis Correlation for the Terminal Velocity 62

3.11 Summary 67

3.e Example Problems 70

3.p Practice Problems 110

4 Applications in Heat Transfer 145

4.1 Introduction 145

4.2 Steady-State Heat Transfer with End Effects 146

4.3 Film and Penetration Theory Approximations 153

4.4 Small Biot Number Approximation 159

4.5 Small Peclet Number Approximation 163

4.6 Boundary-Layer or Large Peclet Number Approximation 167

4.7 Heat Transfer with Phase Change 173

4.8 Temperature-Dependent Physical Properties 180

4.9 Thermally Driven Free Convection: Boussinesq Approximation 183

4.10 Dimensional Analysis Correlation for Cooking a Turkey 187

4.11 Summary 193

4.e Example Problems 196

4.p Practice Problems 224

5 Applications in Mass Transfer 252

5.1 Introduction 252

5.2 Film Theory Approximation 253

5.3 Penetration Theory Approximation 259

5.4 Small Peclet Number Approximation 261

5.5 Small Damköhler Number Approximation 266

5.6 Large Peclet Number Approximation 269

5.7 Quasi-Steady-State Approximation 273

5.8 Membrane Permeation with Nonconstant Diffusivity 277

5.9 Solutally Driven Free Convection Due to Evapotranspiration 281

5.10 Dimensional Analysis for a Membrane-Lung Oxygenator 287

5.11 Summary 293

5.e Example Problems 297

5.p Practice Problems 336

6 Applications in Mass Transfer with Chemical Reaction 360

6.1 Introduction 360

6.2 Concept of the Microscale Element 362

6.3 Scaling the Microscale Element 364

6.4 Slow Reaction Regime 371

6.5 Intermediate Reaction Regime 371

6.6 Fast Reaction Regime 372

6.7 Instantaneous Reaction Regime 373

6.8 Scaling the Macroscale Element 377

6.9 Kinetic Domain of the Slow Reaction Regime 380

6.10 Diffusional Domain of the Slow Reaction Regime 381

6.11 Implications of Scaling Analysis for Reactor Design 381

6.12 Mass-Transfer Coefficients for Reacting Systems 387

6.13 Design of a Continuous Stirred Tank Reactor 390

6.14 Design of a Packed Column Absorber 394

6.15 Summary 397

6.p Practice Problems 399

7 Applications in Process Design 414

7.1 Introduction 414

7.2 Design of a Membrane Lung Oxygenator 415

7.3 Pulsed Single-Bed Pressure-Swing Adsorption 424

7.4 Thermally Induced Phase-Separation Process 438

7.5 Fluid-Wall Aerosol Flow Reactor for Hydrogen Production 448

7.6 Summary 464

7.p Practice Problems 467

Appendix A Sign Convention for the Force on a Fluid Particle 480

Appendix B Generalized Form of the Transport Equations 482

B. 1 Continuity Equation 482

B. 2 Equations of Motion 482

B. 3 Equations of Motion for Porous Media 483

B. 4 Thermal Energy Equation 483

B. 5 Equation of Continuity for a Binary Mixture 484

Appendix c Continuity Equation 486

C. 1 Rectangular Coordinates 486

C. 2 Cylindrical Coordinates 487

C. 3 Spherical Coordinates 487

Appendix d Equations of Motion 489

D. 1 Rectangular Coordinates 489

D. 2 Cylindrical Coordinates 490

D. 3 Spherical Coordinates 492

Appendix E Equations of Motion for Porous Media 494

E. 1 Rectangular Coordinates 494

E. 2 Cylindrical Coordinates 494

E. 3 Spherical Coordinates 495

Appendix F Thermal Energy Equation 496

F. 1 Rectangular Coordinates 496

F. 2 Cylindrical Coordinates 497

F. 3 Spherical Coordinates 497

Appendix G Equation of Continuity for a Binary Mixture 499

G.1 Rectangular Coordinates 499

G. 2 Cylindrical Coordinates 500

G. 3 Spherical Coordinates 502

Appendix H Integral Relationships 504

H.1 Leibnitz Formula for Differentiating an Integral 504

H.2 Gauss Ostrogradskii Divergence Theorem 504

Notation 506

Index 515

William B. Krantz, PhD, PE, is the Isaac M. Meyer Chair Professor in the Department of Chemical and Biomolecular Engineering at the National University of Singapore, Rieveschl Ohio Eminent Scholar and Professor Emeritus at the University of Cincinnati, and President's Teaching Scholar and Professor Emeritus at the University of Colorado. He is a Fellow of the American Association for the Advancement of Science, American Institute of Chemical Engineers, and the American Society for Engineering Education. He has been the recipient of a Guggenheim and three Fulbright-Hayes fellowships. Dr. Krantz is the editor of three research monographs, the author of over 200 technical papers, and the coinventor on five patents.

Scaling analysis takes the guesswork out of developing and using models

Scaling analysis facilitates assessing the viability of a process or technology without the need for prior bench- or pilot-scale data. It also provides a template for the design of experiments used to explore a new process or to validate a mathematical model. The first comprehensive book to focus on systematic scaling analysis while spanning various disciplines and applications, Scaling Analysis in Modeling Transport and Reaction Processes:

• Provides an overview of the systematic approach to scaling analysis, including the mathematical basis

• Includes detailed chapters that cover specific applications in fluid dynamics, heat transfer, mass transfer, mass transfer with chemical reaction, and process design

• Addresses scaling analysis across scientific disciplines and enhances communication across different research areas of applied science, including biology, chemistry, and physics

• Has sixty-two detailed examples that illustrate the scaling method in various applications, as well as chapter-end problems (165 total) that can be used for independent study or as a class assignment

Invaluable for researchers, scientists, and engineers involved in developing and solving models and in designing experiments, this reference is also a great textbook for courses in chemical or mechanical engineering, heat and mass transfer, transport phenomena, mathematical modeling, unit operations, and fluid dynamics.