Advanced Transport Phenomena: Analysis, Modeling, and Computations

Contents
Preface
Topical outline
Notation
1 Introduction
	1.1 What, why, and how?
		1.1.1 What?
		1.1.2 Why?
		1.1.3 How?
		1.1.4 Conservation statement
		1.1.5 The need for constitutive models
		1.1.6 Common constitutive models
	1.2 Typical transport property values
		1.2.1 Viscosity: pure gases and vapors
		1.2.2 Viscosity: liquids
		1.2.3 Thermal conductivity
		1.2.4 Diffusivity
	1.3 The continuum assumption and the field variables
		1.3.1 Continuum and pointwise representation
		1.3.2 Continuum vs. molecular
		1.3.3 Primary field variables
		1.3.4 Auxiliary variables
	1.4 Coordinate systems and representation of vectors
		1.4.1 Cartesian coordinates
		1.4.2 Cylindrical coordinates
		1.4.3 Spherical coordinates
		1.4.4 Gradient of a scalar field
	1.5 Modeling at various levels
		1.5.1 Levels based on control-volume size
		1.5.2 Multiscale models
		1.5.3 Multiscale modeling below the continuum level
	1.6 Model building: general guidelines
	1.7 An example application: pipe flow and tubular reactor
		1.7.1 Pipe flow: momentum transport
		1.7.2 Laminar or turbulent?
		1.7.3 Use of dimensionless numbers
		1.7.4 Pipe flow: heat transport
		1.7.5 Pipe flow: mass exchanger
		1.7.6 Pipe flow: chemical reactor
	1.8 The link between transport properties and molecular models
		1.8.1 Kinetic theory concepts
		1.8.2 Liquids
		1.8.3 Transport properties of solids
	1.9 Six decades of transport phenomena
	1.10 Closure
	Summary
	Additional Reading
	Problems
2 Examples of transport and system models
	2.1 Macroscopic mass balance
		2.1.1 Species balance equation
		2.1.2 Transient balance: tracer studies
		2.1.3 Overall mass balance
	2.2 Compartmental models
		2.2.1 Model equations
		2.2.2 Matrix representation
		2.2.3 A numerical IVP solver in MATLAB
	2.3 Macroscopic momentum balance
		2.3.1 Linear momentum
		2.3.2 Angular momentum
	2.4 Macroscopic energy balances
		2.4.1 Single inlet and outlet
		2.4.2 The Bernoulli equation
		2.4.3 Sonic and subsonic flows
		2.4.4 Cooling of a solid: a lumped model
	2.5 Examples of differential balances: Cartesian
		2.5.1 Heat transfer with nuclear fission in a slab
		2.5.2 Mass transfer with reaction in a porous catalyst
		2.5.3 Momentum transfer: unidirectional flow in a channel
	2.6 Examples of differential models: cylindrical coordinates
		2.6.1 Heat transfer with generation
		2.6.2 Mass transfer with reaction
		2.6.3 Flow in a pipe
	2.7 Spherical coordinates
	2.8 Examples of mesoscopic models
		2.8.1 Tubular reactor with heat transfer
		2.8.2 Heat transfer in a pin fin
		2.8.3 Countercurrent heat exchanger
		2.8.4 Counterflow: matrix method
	Summary
	Problems
3 Flow kinematics
	3.1 Eulerian description of velocity
	3.2 Lagrangian description: the fluid particle
	3.3 Acceleration of a fluid particle
	3.4 The substantial derivative
	3.5 Dilatation of a fluid particle
	3.6 Mass continuity
	3.7 The Reynolds transport theorem
	3.8 Vorticity and rotation
		3.8.1 Curl in other coordinate systems
		3.8.2 Circulation along a closed curve
	3.9 Vector potential representation
	3.10 Streamfunctions
		3.10.1 Two-dimensional flows: Cartesian
		3.10.2 Two-dimensional flows: polar
		3.10.3 Streamfunctions in axisymmetric flows
		3.10.4 The relation to vorticity: the E2 operator
	3.11 The gradient of velocity
	3.12 Deformation and rate of strain
		3.12.1 The physical meaning of the rate of strain
		3.12.2 Rate of strain: cylindrical
		3.12.3 Rate of strain: spherical
		3.12.4 Invariants of a tensor
	3.13 Index notation for vectors and tensors
	Summary
	Problems
4 Forces and their representations
	4.1 Forces on fluids and their representation
		4.1.1 Pressure forces
		4.1.2 Viscous forces
		4.1.3 The divergence of a tensor
	4.2 The equation of hydrostatics
		4.2.1 Archimedes’ principle
		4.2.2 The force on a submerged surface: no curvature
		4.2.3 Force on a curved surface
	4.3 Hydrostatics at interfaces
		4.3.1 The nature of interfacial forces
		4.3.2 Contact angle and capillarity
		4.3.3 The Laplace–Young equation
	4.4 Drag and lift forces
	Summary
	Problems
5 Equations of motion and the Navier–Stokes equation
	5.1 Equation of motion: the stress form
		5.1.1 The Lagrangian point particle
		5.1.2 The Lagrangian control volume
		5.1.3 The Eulerian control volume
	5.2 Types of fluid behavior
		5.2.1 Types and classification of fluid behavior
		5.2.2 Stress relations for a Newtonian fluid
	5.3 The Navier–Stokes equation
		5.3.1 The Laplacian of velocity
		5.3.2 Common boundary conditions for flow problems
	5.4 The dimensionless form of the flow equation
		5.4.1 Key dimensionless groups
		5.4.2 The Stokes equation: slow flow or viscous flow
		5.4.3 The Euler equation
	5.5 Use of similarity for scaleup
	5.6 Alternative representations for the Navier–Stokes equations
		5.6.1 Plane flow: the vorticity–streamfunction form
		5.6.2 Plane flow: the streamfunction representation
		5.6.3 Inviscid and potential flow
		5.6.4 The velocity–vorticity formulation
		5.6.5 Slow flow in terms of vorticity
		5.6.6 The pressure Poisson equation
	5.7 Constitutive models for non-Newtonian fluids
	Summary
	Problems
6 Illustrative flow problems
	6.1 Introduction
		6.1.1 Summary of equations
		6.1.2 Simplifications
		6.1.3 Solution methods
	6.2 Channel flow
		6.2.1 Entry-region flow in channels or pipes
		6.2.2 General solution
		6.2.3 Pressure-driven flow
		6.2.4 Shear-driven flow
		6.2.5 Gravity-driven flow
	6.3 Axial flow in cylindrical geometry
		6.3.1 Circular pipe
		6.3.2 Annular pipe: pressure-driven
		6.3.3 Annular pipe: shear-driven
	6.4 Torsional flow
	6.5 Radial flow
	6.6 Flow in a spherical gap
	6.7 Non-circular channels
	6.8 The lubrication approximation
		6.8.1 Flow between two inclined plates
		6.8.2 Flow in a tapered pipe
	6.9 External flow
	6.10 Non-Newtonian viscoinelastic fluids
		6.10.1 A power-law model
		6.10.2 Flow of a Bingham fluid in a pipe
		6.10.3 The Rabinowitsch equation
	6.11 The effect of fluid elasticity
	6.12 A simple magnetohydrodynamic problem
	Summary
	Additional Reading
	Problems
7 The energy balance equation
	7.1 Application of the first law of thermodynamics to a moving control volume
	7.2 The working rate of the forces
	7.3 Kinetic energy and internal energy equations
	7.4 The enthalpy form
	7.5 The temperature equation
	7.6 Common boundary conditions
	7.7 The dimensionless form of the heat equation
	7.8 From differential to macroscopic
	7.9 Entropy balance and the second law of thermodynamics
		7.9.1 Some definitions from thermodynamics
	Summary
	Problems
8 Illustrative heat transport problems
	8.1 Steady heat conduction and no generation
		8.1.1 Constant conductivity
		8.1.2 Variable thermal conductivity
		8.1.3 Two-dimensional heat conduction problems
	8.2 Heat conduction with generation: the Poisson equation
		8.2.1 The constant-generation case
	8.3 Conduction with temperature-dependent generation
		8.3.1 Linear variation with temperature
		8.3.2 Non-linear variation with temperature
		8.3.3 Two-dimensional Poisson problems
	8.4 Convection effects
		8.4.1 Transpiration cooling
		8.4.2 Convection in boundary layers
	8.5 Mesoscopic models
		8.5.1 Heat transfer from a fin
		8.5.2 A single-stream heat exchanger
	8.6 Volume averaging or lumping
		8.6.1 Cooling of a sphere in a liquid
		8.6.2 An improved lumped model
	Summary
	Problems
9 Equations of mass transfer
	9.1 Preliminaries
	9.2 Concentration jumps at interfaces
	9.3 The frame of reference and Fick’s law
	9.4 Equations of mass transfer
		9.4.1 Mass basis
		9.4.2 Mole basis
		9.4.3 Boundary conditions
	9.5 From differential to macroscopic
	9.6 Complexities in diffusion
	Summary
	Problems
10 Illustrative mass transfer problems
	10.1 Steady-state diffusion: no reaction
		10.1.1 Summary of equations
	10.2 The film concept in mass-transfer analysis
		10.2.1 Fluid–solid interfaces
		10.2.2 Gas–liquid interfaces: the two-film model
	10.3 Mass transfer with surface reaction
		10.3.1 Heterogeneous reactions: the film model
	10.4 Mass transfer with homogeneous reactions
		10.4.1 Diffusion in porous media
		10.4.2 Diffusion and reaction in a porous catalyst
		10.4.3 First-order reaction
		10.4.4 Zeroth-order reaction
		10.4.5 Transport in tissues: the Krogh model
		10.4.6 mth-order reaction
	10.5 Models for gas–liquid reaction
		10.5.1 Analysis for the pseudo-first-order case
		10.5.2 Analysis for instantaneous asymptote
		10.5.3 The second-order case: an approximate solution
		10.5.4 The instantaneous case: the effect of gas film resistance
	10.6 Transport across membranes
		10.6.1 Gas transport: permeability
		10.6.2 Complexities in membrane transport
		10.6.3 Liquid-separation membranes
	10.7 Transport in semi-permeable membranes
		10.7.1 Reverse osmosis
		10.7.2 Concentration-polarization effects
		10.7.3 The Kedem–Katchalsky model
		10.7.4 Transport in biological membranes
	10.8 Reactive membranes and facilitated transport
		10.8.1 Reactive membrane: facilitated transport
		10.8.2 Co- and counter-transport
	10.9 A boundary-value solver in MATLAB
		10.9.1 Code-usage procedure
		10.9.2 BVP4C example: the selectivity of a catalyst
	Summary
	Additional Reading
	Problems
11 Analysis and solution of transient transport processes
	11.1 Transient conduction problems in one dimension
	11.2 Separation of variables: the slab with Dirichlet conditions
		11.2.1 Slab: temperature profiles
		11.2.2 Slab: heat flux
		11.2.3 Average temperature
	11.3 Solutions for Robin conditions: slab geometry
	11.4 Robin case: solutions for cylinder and sphere
	11.5 Two-dimensional problems: method of product solution
	11.6 Transient non-homogeneous problems
		11.6.1 Subtracting the steady-state solution
		11.6.2 Use of asymptotic solution
	11.7 Semi-infinite-slab analysis
		11.7.1 Constant surface temperature
		11.7.2 Constant flux and other boundary conditions
	11.8 The integral method of solution
	11.9 Transient mass diffusion
		11.9.1 Constant diffusivity model
		11.9.2 The penetration theory of mass transfer
		11.9.3 The effect of chemical reaction
		11.9.4 Variable diffusivity
	11.10 Periodic processes
		11.10.1 Analysis for a semi-infinite slab
		11.10.2 Analysis for a finite slab
	11.11 Transient flow problems
		11.11.1 Start-up of channel flow
		11.11.2 Transient flow in a semi-infinite mass of fluid
		11.11.3 Flow caused by an oscillating plate
		11.11.4 Start-up of Poiseuille flow
		11.11.5 Pulsatile flow in a pipe
	11.12 A PDE solver in MATLAB
		11.12.1 Code usage
		11.12.2 Example general code for 1D transient conduction
	Summary
	Additional Reading
	Problems
12 Convective heat and mass transfer
	12.1 Heat transfer in laminar flow
		12.1.1 Preliminaries and the model equations
		12.1.2 The constant-wall-temperature case: the Graetz problem
		12.1.3 The constant-flux case
	12.2 Entry-region analysis
		12.2.1 The constant-wall-temperature case
		12.2.2 The constant-flux case
	12.3 Mass transfer in film flow
		12.3.1 Solid dissolution at a wall in film flow
		12.3.2 Gas absorption from interfaces in film flow
	12.4 Laminar-flow reactors
		12.4.1 A 2D model and key dimensionless groups
		12.4.2 The pure convection model
	12.5 Laminar-flow reactor: a mesoscopic model
		12.5.1 Averaging and the concept of dispersion
		12.5.2 Non-linear reactions
	12.6 Numerical study examples with PDEPE
		12.6.1 The Graetz problem
	Summary
	Problems
13 Coupled transport problems
	13.1 Modes of coupling
		13.1.1 One-way coupling
		13.1.2 Two-way coupling
	13.2 Natural convection problems
		13.2.1 Natural convection between two vertical plates
		13.2.2 Natural convection over a vertical plate
		13.2.3 Natural convection: concentration effects
	13.3 Heat transfer due to viscous dissipation
		13.3.1 Viscous dissipation in plane Couette flow
		13.3.2 Laminar heat transfer with dissipation: the Brinkman problem
	13.4 Laminar heat transfer: the effect of viscosity variations
	13.5 Simultaneous heat and mass transfer: evaporation
		13.5.1 Dry- and wet-bulb temperatures
		13.5.2 Evaporative or sweat cooling
	13.6 Simultaneous heat and mass transfer: condensation
		13.6.1 Condensation of a vapor in the presence of a non-condensible gas
		13.6.2 Fog formation
		13.6.3 Condensation of a binary gas mixture
	13.7 Temperature effects in a porous catalyst
	Summary
	Additional Reading
	Problems
14 Scaling and perturbation analysis
	14.1 Dimensionless analysis revisited
		14.1.1 The method of matrix transformation
		14.1.2 Momentum problems
		14.1.3 Energy transfer problems
		14.1.4 Mass transfer problems
		14.1.5 Example: scaleup of agitated vessels
		14.1.6 Example: pump performance correlation
	14.2 Scaling analysis
		14.2.1 Transient diffusion in a semi-infinite region
		14.2.2 Example: gas absorption with reaction
		14.2.3 Kolmogorov scales for turbulence: an example of scaling
		14.2.4 Scaling analysis of flow in a boundary layer
		14.2.5 Flow over a rotating disk
	14.3 Perturbation methods
		14.3.1 Regular perturbation
		14.3.2 The singular perturbation method
		14.3.3 Example: catalyst with spatially varying activity
		14.3.4 Example: gas absorption with reversible reaction
		14.3.5 Stokes flow past a sphere: the Whitehead paradox
	14.4 Domain perturbation methods
	Summary
	Additional Reading
	Problems
15 More flow analysis
	15.1 Low-Reynolds-number (Stokes) flows
		15.1.1 Properties of Stokes flow
	15.2 The mathematics of Stokes flow
		15.2.1 General solutions: spherical coordinates
		15.2.2 Flow past a sphere: use of the general solution
		15.2.3 Bubbles and drops
		15.2.4 Oseen’s improvement
		15.2.5 Viscosity of suspensions
		15.2.6 Nanoparticles: molecular effects
	15.3 Inviscid and irrotational flow
		15.3.1 Properties of irrotational flow
		15.3.2 The Bernoulli equation revisited
	15.4 Numerics of irrotational flow
		15.4.1 Boundary conditions
		15.4.2 Solutions using harmonic functions
		15.4.3 Solution using singularities
	15.5 Flow in boundary layers
		15.5.1 Relation to the vorticity transport equation
		15.5.2 Flat plate: integral balance
		15.5.3 The integral method: the von Kármán method
		15.5.4 The average value of drag
		15.5.5 Non-flat systems: the effect of a pressure gradient
	15.6 Use of similarity variables
		15.6.1 A simple computational scheme
		15.6.2 Wedge flow: the Falkner–Skan equation
		15.6.3 Blasius flow
		15.6.4 Stagnation-point (Hiemenz) flow
	15.7 Flow over a rotating disk
	Summary: Stokes flow
	Summary: potential flow
	Summary: boundary-layer theory
	Additional Reading
	Problems
16 Bifurcation and stability analysis
	16.1 Introduction to dynamical systems
		16.1.1 Arc-length continuation: a single-equation example
		16.1.2 The arc-length method: multiple equations
	16.2 Bifurcation and multiplicity of DPSs
		16.2.1 A bifurcation example: the Frank-Kamenetskii equation
		16.2.2 Bifurcation: porous catalyst
	16.3 Flow-stability analysis
		16.3.1 Evolution equations and linearized form
		16.3.2 Normal-mode analysis
	16.4 Stability of shear flows
		16.4.1 The Orr–Sommerfeld equation
		16.4.2 Stability of shear layers: the role of viscosity
		16.4.3 The Rayleigh equation
		16.4.4 Computational methods
	16.5 More examples of flow instability
		16.5.1 Kelvin–Helmholtz instability
		16.5.2 Rayleigh–Taylor instability
		16.5.3 Thermal instability: the Bénard problem
		16.5.4 Marangoni instability
		16.5.5 Non-Newtonian fluids
	Summary
	Additional Reading
	Problems
17 Turbulent-flow analysis
	17.1 Flow transition and properties of turbulent flow
	17.2 Time averaging
	17.3 Turbulent heat and mass transfer
	17.4 Closure models
	17.5 Flow between two parallel plates
	17.6 Pipe flow
		17.6.1 The effect of roughness
	17.7 Turbulent boundary layers
	17.8 Other closure models
		17.8.1 The two-equation model: the k−ǫ model
		17.8.2 Reynolds-stress models
		17.8.3 Large-eddy simulation
		17.8.4 Direct numerical simulation
	17.9 Isotropy, correlation functions, and the energy spectrum
	17.10 Kolmogorov’s energy cascade
		17.10.1 Correlation in the spectral scale
	Summary
	Additional Reading
	Problems
18 More convective heat transfer
	18.1 Heat transport in laminar boundary layers
		18.1.1 Problem statement and the differential equation
		18.1.2 The thermal boundary layer: scaling analysis
		18.1.3 The heat integral equation
		18.1.4 Thermal boundary layers: similarity solution
	18.2 Turbulent heat transfer in channels and pipes
		18.2.1 Pipe flow: the Stanton number
	18.3 Heat transfer in complex geometries
	18.4 Natural convection on a vertical plate
		18.4.1 Natural convection: computations
	18.5 Boiling systems
		18.5.1 Pool boiling
		18.5.2 Nucleate boiling
	18.6 Condensation problems
	18.7 Phase-change problems
	Summary
	Additional reading
	Problems
19 Radiation heat transfer
	19.1 Properties of radiation
	19.2 Absorption, emission, and the black body
	19.3 Interaction between black surfaces
	19.4 Gray surfaces: radiosity
	19.5 Calculations of heat loss from gray surfaces
	19.6 Radiation in absorbing media
	Summary
	Additional Reading
	Problems
20 More convective mass transfer
	20.1 Mass transfer in laminar boundary layers
		20.1.1 The low-flux assumption
		20.1.2 Dimensional analysis
		20.1.3 Scaling analysis
		20.1.4 The low-flux case: integral analysis
		20.1.5 The low-flux case: exact analysis
	20.2 Mass transfer: the high-flux case
		20.2.1 The film model revisited
		20.2.2 The high-flux case: the integral-balance model
		20.2.3 The high-flux case: the similarity-solution method
	20.3 Mass transfer in turbulent boundary layers
	20.4 Mass transfer at gas–liquid interfaces
		20.4.1 Turbulent films
		20.4.2 Single bubbles
		20.4.3 Bubble swarms
	20.5 Taylor dispersion
	Summary
	Additional Reading
	Problems
21 Mass transfer: multicomponent systems
	21.1 A constitutive model for multicomponent transport
		21.1.1 Stefan–Maxwell models
		21.1.2 Generalization
	21.2 Non-reacting systems and heterogeneous reactions
		21.2.1 Evaporation in a ternary mixture
		21.2.2 Evaporation of a binary liquid mixture
		21.2.3 Ternary systems with heterogeneous reactions
	21.3 Application to homogeneous reactions
		21.3.1 Multicomponent diffusion in a porous catalyst
		21.3.2 MATLAB implementation
	21.4 Diffusion-matrix-based methods
	21.5 An example of pressure diffusion
	21.6 An example of thermal diffusion
	Summary
	Additional Reading
	Problems
22 Mass transport in charged systems
	22.1 Transport of charged species: preliminaries
		22.1.1 Mobility and diffusivity
		22.1.2 The Nernst–Planck equation
		22.1.3 Potential field and charge neutrality
	22.2 Electrolyte transport across uncharged membranes
	22.3 Electrolyte transport in charged membranes
	22.4 Transport effects in electrodialysis
	22.5 Departure from electroneutrality
	22.6 Electro-osmosis
	22.7 The streaming potential
	22.8 The sedimentation potential
	22.9 Electrophoresis
	22.10 Transport in ionized gases
	Summary
	Additional Reading
	Problems
	Closure
References
Index