Transport Phenomena in Multiphase Flows 2nd Edition

This textbook provides a thorough presentation of the phenomena related to the transport of mass (with and without electric charge), momentum and energy. It lays all the basic physical principles, and then for the more advanced readers, it offers an in-depth treatment with advanced mathematical derivations and ends with some useful applications of the models and equations in specific settings. The important idea behind the book is to unify all types of transport phenomena, describing them within a common framework in terms of cause and effect, respectively, represented by the driving force and the flux of the transported quantity. The approach and presentation are original in that the book starts with a general description of transport processes, providing the macroscopic balance relations of fluid dynamics and heat and mass transfer, before diving into the mathematical realm of continuum mechanics to derive the microscopic governing equations at the microscopic level. The book is a modular teaching tool and is used either for an introductory or for an advanced graduate course. The last six chapters are of interest to more advanced researchers who might be interested in applications in physics, mechanical engineering or biomedical engineering. In particular, this second edition of the book includes two chapters about electric migration, that is the transport of mass that takes place in a mixture under the action of electro-magnetic fields. Electric migration finds many applications in the modeling of energy storage devices, such as batteries and fuel cells. All chapters are complemented with solved exercises that are essential to complete the learning process. Preface to the Second Edition Preface to the First Edition Acknowledgements Contents 1 Thermodynamics and Evolution 1.1 Introduction 1.2 Local Equilibrium 1.3 Introduction to Continuum Mechanics 1.4 Convection and Diffusion 1.5 Viscosity 1.6 Thermal Conductivity 1.7 Molecular Diffusivity 1.8 Molecular Diffusion as an Example of Random Walk 1.9 Examples of Diffusive Processes 1.10 Problems 2 Statics of Fluids 2.1 Hydrostatic Equilibrium 2.2 Manometers 2.3 Surface Tension 2.4 The Young–Laplace Equation 2.5 Contact Angle 2.6 Problems 3 General Features of Fluid Mechanics 3.1 Introduction 3.2 The Reynolds Number 3.3 Boundary Layer and Viscous Resistance 3.4 Boundary Conditions 3.5 Turbulence 3.6 Problems 4 Macroscopic Balances 4.1 Mass Balance and Continuity Equation 4.2 Mechanical Energy Balance and Bernoulli’s Equation 4.3 Momentum Balance 4.4 Recapitulation of the Bernoulli Equation 4.5 Pressure Drops in Pipe Flow 4.6 Localized Pressure Drops 4.7 Flow Around a Submerged Object 4.8 Problems 5 Laminar Flow Fields 5.1 Fully Developed Flow of a Newtonian Fluid in a Pipe 5.2 Fluid Rheology 5.3 Flow of Non-Newtonian Fluids in Circular Pipes 5.4 Flow in Porous Media 5.5 Quasi-Steady Flow 5.6 Capillary Flow 5.7 Problems 6 The Governing Equations of a Simple Fluid 6.1 General Microscopic Balance Equation 6.2 Mass Balance: The Continuity Equation 6.3 Momentum Balance: Cauchy’s Equation 6.4 Angular Momentum Balance 6.5 The Constitutive Equation for Newtonian Fluids 6.6 Energy Balance 6.7 Governing Equations for Incompressible Newtonian Fluids 6.8 The Entropy Equation 7 Laminar Unidirectional Flows 7.1 Flow in Pipes and Channels 7.2 Parallel Plates Viscometer 7.3 Radial Flux Between Two Parallel Disks 7.4 Fluid Flow Due to the Rapid Movement of a Wall 7.5 Lubrication Approximation 7.6 Drainage of a Liquid Film from a Vertical Plate 7.7 Integral Methods 7.8 Problems 8 Laminar Boundary Layer 8.1 Scaling of the Problem 8.2 Blasius Self-similar Solution 8.3 Flow Separation 8.4 von Karman-Pohlhausen Method 8.5 Problems 9 Heat Conduction 9.1 Introduction to Heat Transport 9.2 Unidirectional Heat Conduction 9.3 The Composite Solid 9.4 Quasi-Steady State Approximation 9.5 Problems 10 Conduction with Heat Sources 10.1 Uniform Heat Generation 10.2 Heat Conduction with Chemical Reaction 10.3 Regular Asymptotic Expansion for Small Da 10.4 Singular Asymptotic Expansion for Large Da 10.5 Problems 11 Macroscopic Energy Balance 11.1 Introduction 11.2 The Heat Transfer Coefficient 11.3 Heat Exchangers 11.4 Heat Exchanging Fins 11.5 Problems 12 Time-Dependent Heat Conduction 12.1 Heat Balance Equation 12.2 Heat Conduction in a Semi-infinite Slab 12.3 Temperature Field Generated by a Heat Pulse 12.4 Heat Conduction in a Finite Slab 12.5 Heat Exchange in a Pipe 12.6 Heat Transfer Coefficient in Laminar Flow 12.7 Problems 13 Convective Heat Transport 13.1 Dimensional Analysis of the Problem 13.2 Laminar Thermal Boundary Layer 13.3 Colburn-Chilton Analogy 13.4 The Relation Between δe δT 13.5 Problems 14 Constitutive Equations for Transport of Chemical Species 14.1 Fluxes and Velocities 14.2 Material Balance Equations 14.3 The Constitutive Equations of the Material Fluxes 14.4 Boundary Conditions 14.5 Answers to Some Questions on Material Transport 15 Stationary Material Transport 15.1 Diffusion Through a Stagnant Film 15.2 Diffusion with Heterogeneous Chemical Reaction 15.3 Diffusion with Homogeneous, First-Order Chemical Reaction 15.4 Diffusion with Homogeneous, Second-Order Chemical Reaction 15.5 Problems 16 Non-stationary Material Transport 16.1 Transport Across a Membrane 16.2 Evaporation of a Liquid from a Reservoir 16.3 Slow Combustion of a Coal Particle 16.4 Unsteady Evaporation 16.5 Problems 17 Convective Material Transport 17.1 Mass Transport Through a Fixed Bed 17.2 Laminar Material Boundary Layer 17.3 Mass Boundary Layer for Small Reynolds Number 17.4 Integral Methods 17.5 Quasi Steady State (QSS) Approximation 17.6 Problems 18 Transport Phenomena in Turbulent Flows 18.1 Fundamental Characteristics of Turbulence 18.2 Time- and Length-Scales in Turbulence 18.3 Reynolds-Averaged Equations 18.4 Turbulent Diffusion 18.5 Logarithmic Velocity Profile 18.6 More Complex Models 19 Free Convection 19.1 The Boussinesq Approximation 19.2 Free Convection in a Vertical Channel 19.3 Dimensional Analysis 19.4 The Boundary Layer in Free Convection 19.5 Experimental Correlations 19.6 Heat Transfer with Phase Transition 19.7 Problems 20 Radiant Heat Transfer 20.1 The Law of Stefan-Boltzmann 20.2 Emissivity and Absorptance 20.3 Radiation and Conduction 20.4 Example: The Design of a Solar Panel 20.5 Problems 20.6 Appendix: The Quantum Theory 21 Antidiffusion 21.1 The Chemical Potential 21.2 Chemical Stability 21.3 The Critical Point 21.4 Example: Binary Symmetric Mixtures 21.5 Molecular Diffusion in Binary Symmetric Mixtures 21.6 Non-symmetric Mixtures 21.7 Osmotic Flow 22 Stationary Diffusion 22.1 Harmonic Functions 22.2 Creeping Flow 23 Transport of Electric Charges in Electrolytes 23.1 Thermodynamics of a Mixture with an External Force 23.2 Transport of Electric Charges in Electrolyte Solutions 23.3 Binary Electrolyte Solutions Appendix A Properties of Pure Components at 1 atm Appendix B Viscosity and Surface Tension of Selected Fluids Appendix C Conversion Factors Appendix D Governing Equations D.1 Cartesian Coordinates x, y, z D.2 Cylindrical Coordinates r, φ, z D.3 Spherical Coordinates r, θ, φ Appendix E The Balance Equations (Eulerian Approach) E.1 Conservation of Mass (For One-Phase, One-Component Fluid) E.2 Conservation of Momentum (For Newtonian Incompressible Fluids) E.3 Conservation of Energy (Heat Equation for Incompressible Fluids) E.4 Conservation of Chemical Species (For Incompressible Fluids with Constant Total Concentration) Appendix F Introduction to Linear Algebra F.1 Tensor and Vector Representation F.2 Vector Differential Operators F.3 Integral Theorems Solutions of the Problems Background Reading Subject Index After a brief introduction (Sect. 1.1) on what transport phenomena consist of, in Sect. 1.2 we describe the relation between thermodynamics and transport phenomena, by defining the condition of local equilibrium. We explain that, under very general conditions, although far from thermal and mechanical equilibrium, we can speak of thermodynamic quantities such as temperature and pressure. This idea is further explored in Sect. 1.3, where the basic concepts of continuum mechanics are briefly sketched. Then, in Sect. 1.4, we show that mass, momentum, and energy can be transported through two fundamentally different modalities, namely convection and diffusion. The former is a time reversible process due to a net movement of the fluid, and the related convective fluxes admit exact analytical expressions. On the other hand, diffusion is intrinsically irreversible, and diffusive fluxes are expressed through so called constitutive relations, that characterize the fluid at the molecular level. In the case of ideal gases, as shown in Sects. 1.5–1.7, diffusion of momentum, energy and mass can be modeled rigorously, leading to Newton’s, Fourier’s, and Fick’s constitutive relations, respectively. The analogy between different transport phenomena is further explored in Sect. 1.8, showing that diffusion can be modeled through a random walk process, so that the mean square displacement of the appropriate tracer of momentum, energy or mass grows linearly with time. Finally, in Sect. 1.9,a few examples of diffusion are presented.