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Overview
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.
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Meet the Author
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 FulbrightHayes fellowships. Dr. Krantz is the editor of three research monographs, the author of over 200 technical papers, and the coinventor on five patents.
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Table of Contents
1. Introduction.
1.1 Motivation for Using Scaling Analysis.
1.2 Organization of this Book.
2. Systematic Method for Scaling Analysis.
2.1 Introduction.
2.2 Mathematical Basis for Scaling Analysis.
2.3 OrderofOne Scaling Analysis.
2.4 The Scaling Alternative for Dimensional Analysis.
2.5 Summary.
3. Applications in Fluid Dynamics.
3.1 Introduction.
3.2 Fully Developed Laminar Flow.
3.3 Creeping and LubricationFlow Approximations.
3.4 BoundaryLayer Flow Approximation.
3.5 QuasiSteadyState Flow Approximation.
3.6 Flows with End and Sidewall Effects.
3.7 Free Surface Flow.
3.8 Porous Media Flow.
3.9 Compressible Fluid Flow.
3.10 Dimensional Analysis Correlation for the Terminal Velocity of a Sphere.
3.11 Summary.
3.E Example Problems.
3.E.1 GravityDriven Laminar Film Flow down a Vertical Wall.
3.E.2 Flow between Two Approaching Parallel Circular Flat Plates.
3.E.3 Design of a Hydraulic Ram.
3.E.4 Rotating Disk Flow.
3.E.5 Entry Region Flow between Parallel Plates.
3.E.6 Rotating Flow in an Annulus with End Effects.
3.E.7 Impulsively Initiated PressureDriven Laminar Tube Flow.
3.E.8 Laminar Cylindrical Jet Flow.
3.E.9 GravityDriven Film Flow over Saturated Porous Media.
3.E.10 Flow in a Hollow Fiber Membrane with Permeation.
3.E.11 Falling Head Method for Determining Soil Permeability.
3.P Practice Problems.
3.P.1 Alternate Scales for Laminar Flow between Stationary and Moving Parallel Plates.
3.P.2 Laminar Flow between Stationary and Moving Parallel Plates.
3.P.3 Gravity and PressureDriven Laminar Flow in a Vertical Tube.
3.P.4 Axial Flow in a Rotating Tube.
3.P.5 Laminar Flow between Converging Flat Plates.
3.P.6 Laminar Flow between Diverging Flat Plates.
3.P.7 Laminar Flow in a Diverging Nozzle.
3.P.8 SteadyState Flow between Parallel Circular Disks.
3.P.9 UnsteadyState Flow between Parallel Circular Disks .
3.P.10 SteadyState Flow between Spinning Parallel Circular Disks.
3.P.11 LubricationFlow Approximation for a Hydraulic Ram.
3.P.12 Flow in a Rotating Disk Viscometer.
3.P.13 Flow in an Oscillating Disk Viscometer.
3.P.14 Falling Needle Viscometer.
3.P.15 Leading Edge Considerations for Laminar BoundaryLayer Flow.
3.P.16 Laminar BoundaryLayer Flow with Blowing.
3.P.17 Laminar BoundaryLayer Flow with Suction.
3.P.18 Entry Region Laminar Flow in a Cylindrical Tube.
3.P.19 PressureDriven Flow in an Oscillating Tube.
3.P.20 Countercurrent LiquidGas Flow in a Cylindrical Tube.
3.P.21 Stratified Flow of Two Immiscible Liquid Layers.
3.P.22 Laminar Cylindrical Jet Flow.
3.P.23 Free Surface Flow down a Plane with Condensation.
3.P.24 Free Surface Flow over a Horizontal Filter.
3.P.25 CurtainCoating Flow.
3.P.26 Flow in a SemiInfinite Porous Media Bounded by a Flat Plate.
3.P.27 Porous Media Flow between Parallel Flat Plates.
3.P.28 GravityDriven Film Flow over a Saturated Porous Media.
3.P.29 Radial Flow from a Porous Cylindrical Tube.
3.P.30 EntryRegion Flow in a Tube with a Porous Annulus .
3.P.31 SteadyState Laminar Flow of a Compressible Gas..
3.P.32 Velocity Profile Distortion Effects Owing to Fluid Injection and Withdrawal .
3.P.33 Flow between Parallel Impermeable and Permeable Flat Plates.
3.P.34 Flow in an Annulus with Fluid Injection and Withdrawal.
3.P.35 Flow in a ClosedEnd Permeable Hollow Fiber Membrane.
3.P.36 Dimensional Analysis for Flow around a Falling Sphere.
3.P.37 Dimensional Analysis for Impulsively Initiated Laminar Tube Flow.
3.P.38 Dimensional Analysis for Flow in an Oscillating Tube.
3.P.39 Dimensional Analysis for CurtainCoating Flow.
3.P.40 Dimensional Analysis for Flow between Parallel Membranes.
3.P.41 Dimensional Analysis for Flow in a Hollow Fiber Membrane.
4. Applications in Heat Transfer.
4.1 Introduction.
4.2 SteadyState Heat Transfer with End Effects.
4.3 Film Theory and Penetration Theory Approximations.
4.4 Small Biot Number Approximation.
4.5 Small Peclet number approximation.
4.6 BoundaryLayer or Large Peclet Number Approximation.
4.7 Heat Transfer with Phase Change.
4.8 TemperatureDependent Physical Properties.
4.9 Thermally Driven Free Convection  the Boussinesq Approximation.
4.10 Dimensional Analysis Correlation for Cooking a Turkey.
4.11 Summary.
4.E Example Problems.
4.E.1 SteadyState Heat Conduction in a Rectangular Fin.
4.E.2 UnsteadyState Resistance Heating in a Wire.
4.E.3 Convective Heat Transfer with Injection through Permeable Walls.
4.E.4 SteadyState Heat Transfer to Falling Film Flow.
4.E.5 UnsteadyState Heat Transfer from a Sphere at Large Biot Numbers.
4.E.6 Evaporative Cooling of a Liquid Film.
4.E.7 Free convection Heat Transfer Adjacent to a Vertical Heated Flat Plate.
4.E.8 Dimensional Analysis Correlation for Electrical Heat Generation in a Wire.
4.P Practice Problems.
4.P.1 SteadyState Heat Conduction in a Slab with Specified Cooling Flux.
4.P.2 SteadyState Conduction in a Slab with Specified Heat Flux.
4.P.3 SteadyState Heat Conduction in a Rectangular Parallelepiped.
4.P.4 SteadyState Conduction in a Cylinder with Specified Temperatures at its Boundaries.
4.P.5 SteadyState Conduction in an Annulus with Specified Temperatures at its Boundaries.
4.P.6 SteadyState Heat Conduction in a Circular Fin.
4.P.7 UnsteadyState Axial Heat Conduction in a Solid Cylinder.
4.P.8 UnsteadyState Radial Heat Conduction in a Solid Cylinder.
4.P.9 UnsteadyState Radial Heat Cconduction in a Spherical Shell.
4.P.10 SteadyState Conduction in a Cylinder with External Phase Convection.
4.P.11 UnsteadyState Heat Transfer to a Sphere at Small Biot Numbers.
4.P.12 UnsteadyState Heat Transfer in a Solid Sphere.
4.P.13 UnsteadyState Convective Heat Transfer to a Plane Wall.
4.P.14 UnsteadyState Convective Heat Transfer to a Solid Cylinder.
4.P.15 Entrance Effect Limitations in Laminar Slit Flow.
4.P.16 Convective Heat Transfer for Fully Developed Laminar Flow between Heated Parallel Flat Plates.
4.P.17 Entrance Effect Limitations in the Thermal BoundaryLayer Approximation for Falling Film Flow.
4.P.18 Thermal BoundaryLayer Heat Transfer for Fully Developed Laminar Flow between Heated Parallel Flat Plates.
4.P.19 Heat Transfer from a Hot Inviscid Gas to Fully Developed Laminar Falling Film Flow.
4.P.20 Thermal BoundaryLayer Development along a Heated Flat Plate.
4.P.21 Thermal BoundaryLayer Development with an Unheated Entry Region.
4.P.22 Thermal BoundaryLayer Development with Flux Condition.
4.P.23 Thermal BoundaryLayer Development with Suction.
4.P.24 Evaporative Cooling of a Liquid Film with Radiative Heat Transfer.
4.P.25 Melting of Frozen Soil Owing to Constant Radiative Heat Flux.
4.P.26 Melting of Frozen Soil Initially at SubFreezing Temperature.
4.P.27 Freezing of WaterSaturated Soil Initially above its Freezing Temperature.
4.P.28 Freezing of WaterSaturated Soil Overlaid by Snow.
4.P.29 Heat Conduction in a Cylinder with a TemperatureDependent Thermal Diffusivity.
4.P.30 Entry Region Effects for Free Convection Heat Transfer adjacent to a Vertical Heated Flat Plate.
4.P.31 Free Convection from a Heated Vertical Plate with Wall Suction.
4.P.32 Correlation for Temperature in a Slab with Heat Generation.
4.P.33 Correlation for SteadyState Heat Transfer from a Sphere.
4.P.34 Correlation for Hot Wire Anemometer Performance.
4.P.35 Correlation for UnsteadyState Heat Transfer to a Sphere having a TemperatureDependent Thermal Conductivity.
4.P.36 Characterization of Home Freezer Performance.
5. Applications in Mass Transfer.
5.1 Introduction.
5.2 Film Theory Approximation.
5.3 Penetration Theory Approximation.
5.4 Small Peclet Number Approximation for Laminar Flow with Homogeneous Reaction.
5.5 Small Damköhler Number Approximation for Laminar Flow with Heterogeneous Reaction.
5.6 Large Peclet Number Approximation for Mass Transfer in Falling Film Flow.
5.7 QuasiSteadyState Approximation for Mass Transfer Owing to Evaporation .
5.8 Membrane Permeation with a NonConstant Diffusivity.
5.9 Solutally Driven Free Convection Owing to Evapotranspiration from a Vertical Cylinder.
5.10 Dimensional Analysis for a MembraneLung Oxygenator .
5.11 Summary.
5.E Example Problems.
5.E.1 Evaporative Casting of a Polymer Film .
5.E.2 Taylor Dispersion.
5.E.3 Convective Diffusion in Tapered Pore .
5.E.4 Dissolution of a Spherical Capsule.
5.E.5 Mass Transfer to a Rotating Disk—a Uniformly Accessible Surface.
5.E.6 FieldFlow Fractionation.
5.E.7 Mass Transfer in a Membrane Permeation Cell.
5.E.8 Large Damköhler Number Approximation for Laminar Flow with Heterogeneous Reaction.
5.E.9 Small Thiele Modulus Approximation for Mass Transfer in a Hollow Fiber Membrane.
5.E.10 Dimensional Analysis for Oxygen Diffusion into a Spherical Red Blood Cell.
5.P Practice Problems.
5.P.1 Penetration Theory Approximation for Specified EquationofState.
5.P.2 Error Estimate for Penetration Theory Approximation.
5.P.3 Diffusion in a Tapered Pore.
5.P.4 Liquid Evaporation for Short Contact times.
5.P.5 Mass transfer to Film Flow down a Vertical Plane.
5.P.6 Mass Transfer to Film Flow down a Vertical Cylinder.
5.P.7 Mass Transfer with Chemical Reaction for Flow between SemiPermeable Membranes.
5.P.8 Entrance Effect Limitations for Laminar Tube Flow with Fast Heterogeneous Reaction.
5.P.9 Aeration of Water Containing Aerobic Bacteria.
5.P.10 Dissolution of a Spherical Capsule for a Concentrated Solution.
5.P.11 Dissolution of a Spherical Capsule for a Bimolecular Reaction.
5.P.12 Slow Dissolution of a Cylindrical Capsule.
5.P.13 Dissolution of a Cylindrical Capsule in the Fast Reaction Limit.
5.P.14 Diffusional Growth of a Nucleated Water Droplet.
5.P.15 Crystallization from a Supersaturated Liquid.
5.P.16 Growth of a Liquid Droplet via Diffusion and Heterogeneous Nucleation.
5.P.17 Rusting of a Planar Surface.
5.P.18 Mass Transfer in a Hollow Fiber Membrane.
5.P.19 Permeation Accompanied by Membrane Swelling.
5.P.20 Evaporative Polymer Film Casting for Short Contact Times .
5.P.21 Mass Transfer with Homogeneous Chemical Reaction and a ConcentrationDependent Diffusivity.
5.P.22 Mass Transfer in the Annular Region between Fixed and Rotating Cylinders.
5.P.23 Bulk Flow Effects for Free Convection Mass Transfer from a Vertical Cylinder.
5.P.24 Free Convection Mass Transfer adjacent to a Transpiring Vertical Flat Plate.
5.P.25 Correlation for SteadyState Mass Transfer from a Sphere.
5.P.26 MassTransfer Coefficient Correlation for Film Theory.
5.P.27 Correlation for Free Convection Mass Transfer from an Evaporating Liquid .
5.P.28 Correlation for a Tubular Photocatalytic Reactor.
6. Applications in Mass Transfer with Chemical Reaction.
6.1 Introduction.
6.2 Concept of the Microscale Element.
6.3 Scaling the Microscale Element Slow Reaction Regime.
6.4 Intermediate Reaction Regime.
6.5 Fast Reaction Regime.
6.6 Instantaneous Reaction Regime.
6.7 Scaling the Macroscale Element.
6.8 Kinetic Domain of the Slow Reaction Regime.
6.9 Diffusional Domain of the Slow Reaction Regime.
6.10 Implications of Scaling Analysis for Reactor Design.
6.11 MassTransfer Coefficients for Reacting Systems.
6.12 Design of a Continuous Stirred Tank Reactor.
6.13 Design of a Packed Column Absorber.
6.14 Summary.
6.P Practice Problems.
6.P.1 Criterion for Ignoring the GasPhase Resistance to Mass Transfer.
6.P.2 Penetration Theory Model for Slow Reaction Regime.
6.P.3 Correlation for LiquidPhase MassTransfer Coefficient.
6.P.4 Slow Reaction Regime for NonDilute Solutions.
6.P.5 Microscale Element Scaling for a Reversible Unimolecular Reaction.
6.P.6 Microscale Element Scaling for an Irreversible nthOrder Reaction.
6.P.7 Applicability of the QuasiStationary Hypothesis for Physical Absorption.
6.P.8 Effect of Axial Dispersion in the Macroscale Balance for a Packed Absorption Column.
6.P.9 Intermediate Domain of Slow Reaction Regime.
6.P.10 Macroscale Element Scaling for a Reversible Unimolecular Reaction.
6.P.11 Macroscale Element Scaling for an Irreversible nthOrder Reaction.
6.P.12 Implications of Intermediate Reaction Regime for the Microscale Element on the Macroscale Balance.
6.P.13 Comparison of the Intermediate Reaction Regime and the Diffusional Domain of the Slow Reaction Regime.
6.P.14 Discriminating between the Fast Reaction Regime and the Kinetic Domain of the Slow Reaction Regime.
6.P.15 Transition between the Inner and Surface Domains in the Instantaneous Reaction Regime for Gas Absorption.
6.P.16 Fast Reaction Regime for an nthOrder Reaction.
6.P.17 SteadyState Approximation for the Inner Domain of the Instantaneous Reaction Regime.
6.P.18 Improved Model for the Inner Domain of the Instantaneous Reaction Regime.
6.P.19 Model for the MassTransfer Coefficient for a FirstOrder Reversible Reaction in the Intermediate Reaction Regime.
6.P.20 Model for the MassTransfer Coefficient for a FirstOrder Reversible Reaction in the Fast Reaction Regime.
6.P.21 Macroscale Element Scaling for a ZerothOrder Reversible Reaction.
6.P.22 Design of a CSTR for an nthOrder Reversible Reaction Operating in the Slow Reaction Regime.
6.P.23 Determining Kinetic Parameters for a CSTR.
6.P.24 Determining the Interfacial Area per Unit Volume for a CSTR.
6.P.25 Design of a CSTR for a ZerothOrder Irreversible Reaction.
6.P.26 Use of a Packed Column Absorber to Determine Reaction Order.
6.P.27 Cocurrent versus CounterCurrent Flow in a Packed GasAbsorption Column.
6.P.28 Design of a Packed Column Absorber for an nthOrder Irreversible Reaction.
6.P.29 Design of a Packed Gas Absorption Column for a FirstOrder Irreversible Reaction.
7. Applications in Process Design.
7.1. Introduction.
7.2. Design of a MembraneLung Oxygenator.
7.3. Pulsed SingleBed PressureSwing Adsorption for the Production of OxygenEnriched Air.
7.4. The Thermally Induced PhaseSeparation Process for Polymeric Membrane Fabrication.
7.5. FluidWall Aerosol Flow Reactor for Hydrogen Production from Methane.
7.6. Summary.
7.P Practice Problems.
7.P.1. Axial Diffusion Effects in an Oscillated MembraneLung Oxygenator.
7.P.2. Transient Flow Effects in an Oscillated MembraneLung Oxygenator.
7.P.3. Effect of Process Parameters on the Performance of an Oscillated MembraneLung Oxygenator.
7.P.4. Correlation for the Sherwood Number for a MembraneLung Oxygenator without Axial Oscillations.
7.P.5. Wall Effects in Pulsed PSA.
7.P.6. Alternative Boundary Conditions for Pulsed PSA  Specified Superficial Velocity and Downstream Pressure.
7.P.7. Alternative Boundary Conditions for Pulsed PSA  Specified Superficial Velocity and Upstream Pressure.
7.P.8. Pressure and Velocity Dependence of the Axial Dispersion Coefficient for Pulsed PSA.
7.P.9. Scaling the Depressurization Step for Pulsed PSA.
7.P.10. Dimensional Analysis Correlation for BedSize Factor for Pulsed PSA.
7.P.11. Estimation of Time Required for TIPS Membrane Casting.
7.P.12. Scaling of the TIPS Process for Concentrated Casting Solutions.
7.P.13. TIPS Membrane Casting with both Convective and Radiative Heat Transfer to the Ambient Gas Phase.
7.P.14. TIPS Casting on a Cold Boundary with a Constant Heat Flux.
7.P.15. Effect of Heat Loss to the Ambient Gas Phase in TIPS Casting.
7.P.16. Upward and Downward Propagating PhaseSeparation Fronts in TIPS Casting.
7.P.17. Low Biot Number Approximation for TIPS Membrane Casting.
7.P.18. Effect of Convective Heat Transfer due to Densification during TIPS Casting.
7.P.19. Scaling the FluidWall Aerosol Flow Reactor by Balancing Convection and Heat Transfer from Wall in Gas Phase.
7.P.20. Scaling the FluidWall Aerosol Flow Reactor by Balancing Convection and Heat Transfer from Carbon Particles in Gas Phase.
7.P.21. Estimation of the Thermal BoundaryLayer Thickness at Upstream End of the FluidWall Aerosol Flow Reactor.
7.P.22. Effect of CarbonParticle Radius on the Thermal Equilibrium Approximation for the FluidWall Aerosol Flow Reactor.
7.P.23. Effect of the Wall Temperature on the Thermal Equilibrium Approximation for the FluidWall Aerosol Flow Reactor.
7.P.24. Implications of Rapid Thermal Decomposition Kinetics for the FluidWall Aerosol Flow Reactor.
7.P.25. Temperature Required to Achieve Complete Fractional Conversion in the FluidWall Aerosol Flow Reactor.
7.P.26. Transformation of the Dimensional Groups Containing the Temperature Difference for the FluidWall Aerosol Flow Reactor.
7.P.27. Transformation of the Dimensional Groups Containing the Sum of Molar Velocities for the FluidWall Aerosol Flow Reactor.
7.P.28. Reconciling the Estimate of Fractional Conversion with Dimensional Analysis for the FluidWall Aerosol Flow Reactor.
7.P.29. Correlation for the Fractional Conversion when Local Thermal Equilibrium Applies in the FluidWall Aerosol Flow Reactor.
7.P.30. Isolating the GasPhase MassTransfer Coefficient in Dimensional Analysis for the FluidWall Aerosol Flow Reactor.
7.P.31. Isolating the Molar Velocity of Hydrogen in Dimensional Analysis for the FluidWall Aerosol Flow Reactor.
1 Appendices.
A Sign Convection for the Force on a Fluid Particle.
B Generalized Form of the Transport Equations.
B.1 Continuity Equation.
B.2 EquationsofMotion.
B.3 EquationsofMotion for Porous Media.
B.4 Thermal Energy Equation.
B.5 Equation of Continuity for a Binary Mixture.
C Continuity Equation.
C.1 Rectangular Coordinates.
C.2 Cylindrical Coordinates.
C.3 Spherical Coordinates.
D EquationsofMotion.
D.1 Rectangular Coordinates.
D.2 Cylindrical Coordinates.
D.3 Spherical Coordinates.
E Describing Equations for Porous Media.
E.1 Rectangular Coordinates.
E.2 Cylindrical Coordinates.
E.3 Spherical Coordinates.
F Thermal Energy Equation.
F.1 Rectangular Coordinates.
F.2 Cylindrical Coordinates.
F.3 Spherical Coordinates.
G Equation of Continuity for a Binary Mixture.
G.1 Rectangular Coordinates.
G.2 Cylindrical Coordinates.
G.3 Spherical Coordinates.
H Integral Relationships.
H.1 Leibnitz Formula for Differentiating an Integral.
H.2 GaussOstrogradskii Divergence Theorem.
Notation.
Index.