Scaling Analysis in Modeling Transport and Reaction Processes: A Systematic Approach to Model Building and the Art of Approximation

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 Order-of-One 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 Lubrication-Flow Approximations.

3.4 Boundary-Layer Flow Approximation.

3.5 Quasi-Steady-State 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 Gravity-Driven 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 Pressure-Driven Laminar Tube Flow.

3.E.8 Laminar Cylindrical Jet Flow.

3.E.9 Gravity-Driven 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 Pressure-Driven 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 Steady-State Flow between Parallel Circular Disks.

3.P.9 Unsteady-State Flow between Parallel Circular Disks .

3.P.10 Steady-State Flow between Spinning Parallel Circular Disks.

3.P.11 Lubrication-Flow 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 Boundary-Layer Flow.

3.P.16 Laminar Boundary-Layer Flow with Blowing.

3.P.17 Laminar Boundary-Layer Flow with Suction.

3.P.18 Entry Region Laminar Flow in a Cylindrical Tube.

3.P.19 Pressure-Driven Flow in an Oscillating Tube.

3.P.20 Countercurrent Liquid-Gas 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 Curtain-Coating Flow.

3.P.26 Flow in a Semi-Infinite Porous Media Bounded by a Flat Plate.

3.P.27 Porous Media Flow between Parallel Flat Plates.

3.P.28 Gravity-Driven Film Flow over a Saturated Porous Media.

3.P.29 Radial Flow from a Porous Cylindrical Tube.

3.P.30 Entry-Region Flow in a Tube with a Porous Annulus .

3.P.31 Steady-State 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 Closed-End 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 Curtain-Coating 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 Steady-State 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 Boundary-Layer or Large Peclet Number Approximation.

4.7 Heat Transfer with Phase Change.

4.8 Temperature-Dependent 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 Steady-State Heat Conduction in a Rectangular Fin.

4.E.2 Unsteady-State Resistance Heating in a Wire.

4.E.3 Convective Heat Transfer with Injection through Permeable Walls.

4.E.4 Steady-State Heat Transfer to Falling Film Flow.

4.E.5 Unsteady-State 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 Steady-State Heat Conduction in a Slab with Specified Cooling Flux.

4.P.2 Steady-State Conduction in a Slab with Specified Heat Flux.

4.P.3 Steady-State Heat Conduction in a Rectangular Parallelepiped.

4.P.4 Steady-State Conduction in a Cylinder with Specified Temperatures at its Boundaries.

4.P.5 Steady-State Conduction in an Annulus with Specified Temperatures at its Boundaries.

4.P.6 Steady-State Heat Conduction in a Circular Fin.

4.P.7 Unsteady-State Axial Heat Conduction in a Solid Cylinder.

4.P.8 Unsteady-State Radial Heat Conduction in a Solid Cylinder.

4.P.9 Unsteady-State Radial Heat Cconduction in a Spherical Shell.

4.P.10 Steady-State Conduction in a Cylinder with External Phase Convection.

4.P.11 Unsteady-State Heat Transfer to a Sphere at Small Biot Numbers.

4.P.12 Unsteady-State Heat Transfer in a Solid Sphere.

4.P.13 Unsteady-State Convective Heat Transfer to a Plane Wall.

4.P.14 Unsteady-State 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 Boundary-Layer Approximation for Falling Film Flow.

4.P.18 Thermal Boundary-Layer 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 Boundary-Layer Development along a Heated Flat Plate.

4.P.21 Thermal Boundary-Layer Development with an Unheated Entry Region.

4.P.22 Thermal Boundary-Layer Development with Flux Condition.

4.P.23 Thermal Boundary-Layer 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 Sub-Freezing Temperature.

4.P.27 Freezing of Water-Saturated Soil Initially above its Freezing Temperature.

4.P.28 Freezing of Water-Saturated Soil Overlaid by Snow.

4.P.29 Heat Conduction in a Cylinder with a Temperature-Dependent 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 Steady-State Heat Transfer from a Sphere.

4.P.34 Correlation for Hot Wire Anemometer Performance.

4.P.35 Correlation for Unsteady-State Heat Transfer to a Sphere having a Temperature-Dependent 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 Quasi-Steady-State Approximation for Mass Transfer Owing to Evaporation .

5.8 Membrane Permeation with a Non-Constant Diffusivity.

5.9 Solutally Driven Free Convection Owing to Evapotranspiration from a Vertical Cylinder.

5.10 Dimensional Analysis for a Membrane-Lung 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 Field-Flow 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 Equation-of-State.

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 Semi-Permeable 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 Concentration-Dependent 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 Steady-State Mass Transfer from a Sphere.

5.P.26 Mass-Transfer 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 Mass-Transfer 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 Gas-Phase Resistance to Mass Transfer.

6.P.2 Penetration Theory Model for Slow Reaction Regime.

6.P.3 Correlation for Liquid-Phase Mass-Transfer Coefficient.

6.P.4 Slow Reaction Regime for Non-Dilute Solutions.

6.P.5 Microscale Element Scaling for a Reversible Unimolecular Reaction.

6.P.6 Microscale Element Scaling for an Irreversible nth-Order Reaction.

6.P.7 Applicability of the Quasi-Stationary 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 nth-Order 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 nth-Order Reaction.

6.P.17 Steady-State 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 Mass-Transfer Coefficient for a First-Order Reversible Reaction in the Intermediate Reaction Regime.

6.P.20 Model for the Mass-Transfer Coefficient for a First-Order Reversible Reaction in the Fast Reaction Regime.

6.P.21 Macroscale Element Scaling for a Zeroth-Order Reversible Reaction.

6.P.22 Design of a CSTR for an nth-Order 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 Zeroth-Order Irreversible Reaction.

6.P.26 Use of a Packed Column Absorber to Determine Reaction Order.

6.P.27 Co-current versus Counter-Current Flow in a Packed Gas-Absorption Column.

6.P.28 Design of a Packed Column Absorber for an nth-Order Irreversible Reaction.

6.P.29 Design of a Packed Gas Absorption Column for a First-Order Irreversible Reaction.

7. Applications in Process Design.

7.1. Introduction.

7.2. Design of a Membrane-Lung Oxygenator.

7.3. Pulsed Single-Bed Pressure-Swing Adsorption for the Production of Oxygen-Enriched Air.

7.4. The Thermally Induced Phase-Separation Process for Polymeric Membrane Fabrication.

7.5. Fluid-Wall Aerosol Flow Reactor for Hydrogen Production from Methane.

7.6. Summary.

7.P Practice Problems.

7.P.1. Axial Diffusion Effects in an Oscillated Membrane-Lung Oxygenator.

7.P.2. Transient Flow Effects in an Oscillated Membrane-Lung Oxygenator.

7.P.3. Effect of Process Parameters on the Performance of an Oscillated Membrane-Lung Oxygenator.

7.P.4. Correlation for the Sherwood Number for a Membrane-Lung 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 Bed-Size 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 Phase-Separation 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 Fluid-Wall Aerosol Flow Reactor by Balancing Convection and Heat Transfer from Wall in Gas Phase.

7.P.20. Scaling the Fluid-Wall Aerosol Flow Reactor by Balancing Convection and Heat Transfer from Carbon Particles in Gas Phase.

7.P.21. Estimation of the Thermal Boundary-Layer Thickness at Upstream End of the Fluid-Wall Aerosol Flow Reactor.

7.P.22. Effect of Carbon-Particle Radius on the Thermal Equilibrium Approximation for the Fluid-Wall Aerosol Flow Reactor.

7.P.23. Effect of the Wall Temperature on the Thermal Equilibrium Approximation for the Fluid-Wall Aerosol Flow Reactor.

7.P.24. Implications of Rapid Thermal Decomposition Kinetics for the Fluid-Wall Aerosol Flow Reactor.

7.P.25. Temperature Required to Achieve Complete Fractional Conversion in the Fluid-Wall Aerosol Flow Reactor.

7.P.26. Transformation of the Dimensional Groups Containing the Temperature Difference for the Fluid-Wall Aerosol Flow Reactor.

7.P.27. Transformation of the Dimensional Groups Containing the Sum of Molar Velocities for the Fluid-Wall Aerosol Flow Reactor.

7.P.28. Reconciling the Estimate of Fractional Conversion with Dimensional Analysis for the Fluid-Wall Aerosol Flow Reactor.

7.P.29. Correlation for the Fractional Conversion when Local Thermal Equilibrium Applies in the Fluid-Wall Aerosol Flow Reactor.

7.P.30. Isolating the Gas-Phase Mass-Transfer Coefficient in Dimensional Analysis for the Fluid-Wall Aerosol Flow Reactor.

7.P.31. Isolating the Molar Velocity of Hydrogen in Dimensional Analysis for the Fluid-Wall 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 Equations-of-Motion.

B.3 Equations-of-Motion 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 Equations-of-Motion.

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 Gauss-Ostrogradskii Divergence Theorem.

Notation.

Index.