# Transport Processes in Chemically Reacting Flow Systems

### Butterworths Series in Chemical Engineering

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Transport Processes in Chemically Reacting Flow Systems discusses the role, in chemically reacting flow systems, of transport processes—particularly the transport of momentum, energy, and (chemical species) mass in fluids (gases and liquids). The principles developed and often illustrated here for combustion systems are important not only for the rational design and development of engineering equipment (e.g., chemical reactors, heat exchangers, mass exchangers) but also for scientific research involving coupled transport processes and chemical reaction in flow systems. The book begins with an introduction to transport processes in chemically reactive systems. Separate chapters cover momentum, energy, and mass transport. These chapters develop, state, and exploit useful quantitative ""analogies"" between these transport phenomena, including interrelationships that remain valid even in the presence of homogeneous or heterogeneous chemical reactions. A separate chapter covers the use of transport theory in the systematization and generalization of experimental data on chemically reacting systems. The principles and methods discussed are then applied to the preliminary design of a heat exchanger for extracting power from the products of combustion in a stationary (fossil-fuel-fired) power plant. The book has been written in such a way as to be accessible to students and practicing scientists whose background has until now been confined to physical chemistry, classical physics, and/or applied mathematics.

## Table of Contents

List of Primary Figures

List of Primary Tables

Preface

1 Introduction to Transport Processes in Chemically Reactive Systems

Introduction

1.1 Physical Factors Governing Reaction Rates and Pollutant Emission: Examples of Partial or Total "Mixing" Rate Limitations

1.1.1 Flame Spread across IC Engine Cylinder

1.1.2 Gaseous Fuel Jet

1.1.3 Single Fuel Droplet and Fuel Droplet Spray Combustion

1.2 Continuum (vs. Molecular) Viewpoint: Length and Time Scales of Fluid-Dynamic Interest

1.3 Types/Uses of "Control" Volumes

1.4 Notion of Conservation Principles and Their Application to Moving Continua

1.5 Notion of "Constitutive" Laws (and Coefficients) for Particular Substances

1.6 Uses of Conservation/Constitutive Principles in Science and Technology

1.6.1 Inference of Constitutive Laws/Coefficients Based on Analysis and Measurement of Simple ("Canonical") Flow/Transport Situations

1.6.2 Solution of Simpler "Prototype" Problems Illustrating Effects of the Basic Interacting Phenomena

1.6.3 Guide Design of Small-Scale or Full-Scale Experiments, and the Interpretation and Generalization of Experimental Results

1.6.4 Comprehensive Exchanger/Reactor Design Predictions via Computer Modeling

1.6.5 Interpretation of Instrument Measurements Made in the Laboratory or Field

Summary

True/False Questions

References

Bibliography

2 Governing Conservation Principles

Introduction

Approach

2.1 Conservation of Mass

2.1.1 Total Mass Conservation

2.1.2 Individual Species Mass Balance

2.1.3 Individual Chemical Element Conservation

2.2 Conservation of Momentum (Mixture)

2.2.1 Linear Momentum Conservation

2.2.2 Angular Momentum Conservation

2.3 Conservation of Energy (First Law of Thermodynamics)

2.4 "Conservation" of Entropy (Second Law of Thermodynamics)

2.5 Alternative ("Derived") Forms of the Conservation (Balance) Equations

2.5.1 Introduction

2.5.2 Origin of the "Accumulation Rate + Net Convective Outflow Rate" Structure of all Conservation Equations for a Fixed Control Volume

2.5.3 Material Derivative Form of the Conservation PDEs

2.5.4 Alternate Forms of the Energy Conservation PDE

2.5.5 Macroscopic Mechanical Energy Equation (Generalized Bernoulli Equation)

2.5.6 Explicit Form of the Differential Entropy Balance when q'" = 0

2.5.7 Explicit Form of the Conservation PDEs in Alternate Orthogonal Coordinate Systems

2.6 Remarks on Important Generalizations

2.6.1 Moving Control Volumes and "Jump" Conditions across Moving (or Fixed) Discontinuities (Shock Waves, Phase Boundaries, etc.)

2.6.2 Conservation Equations Using an Accelerating (Noninertial) Coordinate Frame

2.6.3 Approach (Reynolds') to Treatment of Turbulence via Time- Averaging the Conservation Equations

2.6.4 Approach to the Treatment of Multiphase Continua via Volume-Averaging the Conservation Equations

2.7 Comments on the Matrix of Fluid Mechanics

2.7.1 Continuum/Molecular

2.7.2 Compressible/Incompressible

2.7.3 Viscous/Inviscid

2.7.4 Newtonian/Non-Newtonian

2.7.5 Steady/Unsteady

2.7.6 Laminar/Turbulent

2.7.7 Multidimensional/One-Dimensional

Summary

True/False Questions

Exercises

References

Bibliography: Conservation Principles

3 Constitutive Laws: The Diffusion Flux Laws and Their Coefficients

3.1 Closure via Constitutive Laws/Coefficients

3.1.1 Equations of State

3.1.2 Chemical Kinetics

3.1.3 Diffusion Flux-Driving Force Laws/Coefficients

3.1.4 General Constraints on the Diffusion Flux Laws

3.2 Linear-Momentum Diffusion (Contact Stress) vs. Rate of Fluid-Parcel Deformation

3.2.1 The "Extra" Stress Operator and Its Components

3.2.2 Stokes' Extra Stress vs. Rate of Deformation Relation

3.2.3 Energy Equation in Terms of the Work Done by the Fluid against the Extra Stress

3.2.4 Viscous Dissipation and Its Consequences

3.2.5 The Dynamic Viscosity Coefficient of Gases and Liquids—Real and Effective

3.3 Energy Diffusion Flux vs. Spatial Gradients of Temperature and Species Concentrat

3.3.1 Fourier's Heat-Flux Law

3.3.2 Species Diffusion Contribution to Energy Flux

3.3.3 Entropy Production and Diffusion Associated with "Fourier" Energy Diffusion

3.3.4 The Thermal Conductivity Coefficient of Gases, Liquids, and Solids—Real and Effective

3.4 Mass Diffusion Flux vs. Spatial Gradients of Composition

3.4.1 Fick's Diffusion-Flux Law for Chemical Species

3.4.2 Corresponding Chemical-Element Diffusion Fluxes

3.4.3 Multicomponent Diffusion Flux Law: Entropy Production and Entropy Diffusion Associated with Chemical Species Diffusion

3.4.4 Solute Diffusivities in Gases, Liquids, and Solids—Real and Effective

3.5 Limitations of Linear Local Flux vs. Local Driving Force Constitutive Laws

3.5.1 "Nonlinear" Fluids

3.5.2 Nonlocal Spatial Behavior

3.5.3 Nonlocal Temporal Behavior—Fluids with Memory

3.5.4 Multiphase Effects: Nonlinear Species "Drag" Laws

Summary

True/False Questions

Exercises

References

Bibliography: Constitutive Laws

4 Momentum Transport Mechanisms; Rates, and Coefficients

4.1 Relevance of Fluid Dynamics and the Classification of Fluid Flow Systems

4.1.1 Role of Fluid Mechanics in the Analysis/Design of Chemical Reactors, Separators, etc.

4.1.2 Criteria for Quiescence

4.1.3 Further Classification of Continuum Fluid Flows

4.1.4 Interactive Role of Experiment and Theory

4.1.5 Overall vs. Local Momentum and Mass Balances

4.2 Mechanisms of Momentum Transport, Their Associated Transport Properties, and Analogies to Energy and Mass Transport

4.2.1 Momentum Convection

4.2.2 Momentum Diffusion

4.2.3 Real and Effective Fluid Viscosities

4.2.4 Analogies to Energy and Mass Transport, and Their Uses

4.3 Convective Momentum Transport in Globally Inviscid Flow

4.3.1 Steady One-Dimensional Compressible Fluid Flow

4.3.2 "Shock" Waves, Sound Waves, Detonation Waves, and "Deflagration" Waves

4.3.3 Remarks on Multidimensional Inviscid Steady Flow

4.4 Velocity Fields and Corresponding Surface Momentum-Transfer Coefficients

4.4.1 Relation between Local Velocity Fields, Wall Momentum Transfer Rates, and Wall Coefficients

4.4.2 Conservation Equations Governing the Velocity and Pressure Fields

4.4.3 Typical Boundary Conditions

4.4.4 Outline of Solution Methods

4.5 Velocity Fields and Surface Momentum-Transport Coefficients: Steady Laminar Flow of an Incompressible Newtonian Fluid

4.5.1 Momentum Transfer from the Fluid Flowing in a Straight Duct of Circular Cross Section

4.5.2 Momentum Diffusion Boundary-Layer Theory: Example of Laminar Flow Past a Flat Plate at Zero Incidence

4.6 Momentum Transfer in "Steady" Turbulent Flows: Entrainment by Jets

4.6.1 Introduction

4.6.2 Laminar Round Jet of an Incompressible Newtonian Fluid: Far- Field

4.6.3 Fully Turbulent Round Jet: Far-Field

4.6.4 Generalization: Turbulence "Modeling"

4.6.5 Entrainment Limitations and Recirculation in Confined Ducts

4.6.6 Concluding Remarks on Turbulent Jet Mixing

4.7 Momentum Transfer for Fluid Flow in Porous Media or Packed Beds

Summary

True/False Questions

Exercises

References

Bibliography: Momentum Transport

5 Energy Transport Mechanisms, Rates, and Coefficients

5.1 Relevance

5.2 Mechanisms of Energy Transport

5.2.1 Convection

5.2.2 Diffusion (Heat Conduction)

5.2.3 Radiation

5.3 Obtaining Temperature Fields and Corresponding Surface Energy-Transfer Rates and Coefficients

5.3.1 Relation between Temperature Field, Wall Heat Transfer Rates, and Heat-Transfer Coefficients

5.3.2 Conservation Equation Governing the Temperature Field, Typical Boundary Conditions, and Outline of Solution Methods

5.4 Temperature Distributions and Surface Heat-Transfer for Quiescent Media of Uniform Composition

5.4.1 Relevance 226

5.4.2 Criteria for Quiescence

5.4.3 Steady-State Heat Conduction

5.4.4 Steady-State Heat Conduction Outside of an Isothermal Sphere

5.4.5 Steady-State, Quasi-One-Dimensional Heat Conduction

5.4.6 Transient Heat Diffusion: Concept of a Thermal Boundary Layer

5.5 Temperature Distributions and Surface Heat-Transfer Coefficients in Steady Laminar Flows

5.5.1 Introduction

5.5.2 Thermal Boundary Layer Adjacent to a Flat Plate

5.5.3 Convective Heat Transfer from/to an Isolated Sphere

5.5.4 Convective Heat Transfer to/from the Fluid Flowing in a Straight Circular Duct (Re < 2 x 103)

5.5.5 Heat Exchange between a Fluid and a Porous Medium or "Packing"

5.5.6 Diffusion from a Steady Point Source in a Uniform Moving

5.6 Time-Averaged Temperature Distributions and Surface Heat-Transfer Coefficients in "Steady" Turbulent Flows

5.6.1 Criteria for Transition to Turbulence and the Effects of Turbulence

5.6.2 Forced Convective Turbulent Heat Transfer from/to Straight, Smooth Ducts

5.6.3 Turbulent Thermal Boundary Layers on a Smooth Flat Plate

5.6.4 Time-Averaged Total Heat Transfer Rate from/to an Isolated Sphere at High Re

5.6.5 Continuous Point Heat Source in a Uniform Turbulent Stream

5.7 Analogies between Energy and Momentum Transport

5.7.1 Fully Turbulent Jet Flow into a Co-Flowing Surrounding Stream

5.7.2 Analogies between Momentum and Energy Transfer for Solid- Wall Boundary-Layer Flows: Extended Reynolds' Analogy for Surface-Transfer Coefficients in Fluids with Pr φ 1

5.7.3 Remarks on Recent Developments in Turbulence Modeling

5.8 Convective Energy Transport in Chemically Reacting Systems

5.9 Remarks on the Implications and Treatment of Radiation-Energy Transfer 2ment of Radiation in High-Temperature Chemical Reactors

5.9.1 Radiation Emission from, and Exchange between Opaque Solid Surfaces

5.9.2 Radiant Emission and Transmission by Dispersed Particulate Matter

5.9.3 Radiant Emission and Transmission by IR-Active Vapors

5.9.4 Remarks on the Quantitative Treatment of Radiation in High-Temperature Chemical Reactors

Summary

Appendix 5.1 Outline of Fourier Method of "Separation of Variables" and Eigenfunction Expansions

True/False Questions

Exercises

References

Bibliography

6 Mass Transport Mechanisms, Rates, and Coefficients

6.1 Relevance

6.1.1 Transport-Controlled Situations

6.1.2 Kinetically Limited Situations

6.1.3 "Ideal" Steady-Flow Chemical Reactors (PFR, WSR)

6.2 Mechanisms of Mass Transport and Their Associated Transport Properties

6.2.1 Convection

6.2.2 Concentration Diffusion (Fick, Brownian, or "Eddy'")

6.2.3 Free-Molecular "Flight"

6.2.4 Solute Diffusivities—Real and Effective

6.2.5 Drift Velocities due to Solute-Applied Forces

6.2.6 Particle (or Heavy Molecule) "Slip," Pressure Diffusion, and Inertial Separation

6.3 Obtaining Concentration Fields and Corresponding Surface-Transfer Rates/Coefficients

6.3.1 Relation between Concentration Fields, Wall Mass-Transfer Rates, and Mass-Transfer Coefficients

6.3.2 Conservation Equations Governing the Concentration Fields, Typical Boundary Conditions, and Outline of Solution Methods

6.3.3 Analogies between Mass and Energy Transfer and the Processes that "Break" These Analogies

6.3.4 Mass-Transfer Coefficient Correction Factors to Account for Analogy-Breaking Phenomena

6.4 Concentration Distributions and Surface Mass-Transfer Coefficients for Quiescent Media

6.4.1 Criteria for Quiescence

6.4.2 Composite Planar Slabs

6.4.3 Stefan-Flow Effects on Mass- and Energy-Transport Coefficients

6.4.4 Steady Mass Diffusion with Simultaneous Chemical Reaction: Catalyst Pellet "Effectiveness Factors"

6.4.5 Transient Mass Diffusion: Concept of a Mass-Transfer (Concentration) Boundary Layer

6.5 Convective Mass Transfer in Laminar- and Turbulent-Flow Systems

6.5.1 Analogies to Energy Transfer

6.5.2 Analogies to Momentum Transport: High Schmidt-Number Effects

6.5.3 Treatment of Mass-Transfer Problems with Chemical Nonequilibrium (Kinetic) Boundary Conditions

6.5.4 Combined Energy and Mass Transport: Recovery of Mainstream Chemical and Kinetic Energy

6.5.5 "Analogies" in the Presence of a Homogeneous Exoergic Chemical Reaction: "Conserved Quantities" in Single-Phase Chemically Reacting Flows

6.6 Remarks on Two-Phase Flow Mass-Transfer Effects of Inertial "Slip" and "Isokinetic" Sampling

6.6.1 Pure Inertial Impaction at Supercritical Stokes' Numbers: Cylinder in Cross-Flow

6.6.2 Effective Diffusivity of Particles in Turbulent Flow

6.6.3 Eddy Impaction

6.7 Residence-Time Distributions: Tracer "Diagnostics" with Application to the Mathematical Modeling of Nonideal-Flow Reactors

6.7.1 Introduction

6.7.2 Ideal Plug-Flow Reactor (PFR) and Its RTDF

6.7.3 Well-Stirred Reactor (WSR) and Its RTDF

6.7.4 RTDF for Composite Systems

6.7.5 Real Reactors as a Network of Ideal Reactors: Modular Modeling

6.7.6 Statistical Microflow (Random Eddy Surface-Renewal) Models of Interfacial Mass Transport in Turbulent-Flow Systems

6.7.7 Extinction, Ignition, and the "Parametric Sensitivity" of Chemical Reactors

Summary

True/False Questions

Exercises

References

Bibliography

7 Similitude Analysis with Application to Chemically Reactive Systems—Overview of the Role of Experiment and Theory

7.1 Introduction and Objectives

7.1.1 Possibility of Scale-Model Testing and the Quantitative Exploitation of Similarity in Generalizing the Results of Experiments or Computations

7.1.2 Types of Similarity 406

7.1.3 Nondimensional Presentations in Science and Engineering

7.2 Dimensional Analysis, Similitude Analysis, Analogies, and Scale-Model Theory

7.2.1 Dimensional Analysis and Its Advantages/Limitations

7.2.2 Similitude Analysis 414

7.2.3 "Partial Modeling" of Chemically Reacting Systems

7.2.4 Alternative Interpretation of Dimensionless Groups ("Eigen- Ratios")

7.3 Concluding Remarks: Mathematical Modeling in Chemically Reacting Systems—An Interactive, Balanced Approach

Summary

Similitude Analysis: Overview of the Role of Experiment and Theory in Reactive Fluid Mechanics

Overview

Review of the Uses of Conservation and Constitutive Laws

Concluding Remarks

True/False Questions

Exercises

Appendix 7.1 Outline of Procedure for Similitude Analysis

References

Bibliography: Similitude/Dimensional Analysis

8 Problem-Solving Techniques, Aids, Philosophy: Forced Convective Heat and Mass Transfer to a Tube in Cross-Flow

Introduction

Tentative Simplifying Assumptions

8.1 Energy-Transfer Rate from the Combustion Products to the Tube

8.1.1 Reynolds' Number and Dimensionless Heat-Transfer Coefficient

8.1.2 Heat-Transfer Rate

8.2 Heat-Exchanger Tube-Fouling Rate Predictions due to Ash Accumulation

8.2.1 Evaluation and Inclusion of the Role of Particle Thermophoresis

8.2.2 Defense of "Single-Phase" Flow Assumption

Exercises and Discussion Questions

References

Appendix 8.1 Recommendations on Problem-Solving

Appendix 8.2 Outline of the Method of Finite Differences (MFD) for the Numerical Solution of Partial Differential (Field) Equations (PDEs) and Ancillary Boundary Conditions (BCs)

Appendix 8.3 Outline of the Method of Finite Elements (MFE) for the Numerical Solution of PDEs on Domains of Complicated Shape

Appendix 8.4 Outline of the Method of Weighted Residuals (MWR) for the Approximate Solution of Partial Differential (Field) Equations and Ancillary Conditions (BCs, ICs)

Appendix 8.5 Physical Constants

Appendix 8.6 Metric System Notes/Conversion Factors

Solutions to Selected Exercises

Index

## Product details

- No. of pages: 568
- Language: English
- Copyright: © Butterworth-Heinemann 1986
- Published: June 24, 1986
- Imprint: Butterworth-Heinemann
- eBook ISBN: 9781483162683

## About the Author

### Daniel E. Rosner

## About the Editor

### Howard Brenner

#### Affiliations and Expertise

Massachusetts Institute of Technology

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