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Fluid Mechanics for Chemical Engineers with Microfluidics and CFD / Edition 2

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Overview

The Chemical Engineer's Practical Guide to Contemporary Fluid Mechanics

Since most chemical processing applications are conducted either partially or totally in the fluid phase, chemical engineers need a strong understanding of fluid mechanics. Such knowledge is especially valuable for solving problems in the biochemical, chemical, energy, fermentation, materials, mining, petroleum, pharmaceuticals, polymer, and waste-processing industries.

Fluid Mechanics for Chemical Engineers, Second Edition, with Microfluidics and CFD, systematically introduces fluid mechanics from the perspective of the chemical engineer who must understand actual physical behavior and solve real-world problems. Building on a first edition that earned Choice Magazine's Outstanding Academic Title award, this edition has been thoroughly updated to reflect the field's latest advances.

This second edition contains extensive new coverage of both microfluidics and computational fluid dynamics, systematically demonstrating CFD through detailed examples using FlowLab and COMSOL Multiphysics. The chapter on turbulence has been extensively revised to address more complex and realistic challenges, including turbulent mixing and recirculating flows.

Part I offers a clear, succinct, easy-to-follow introduction to macroscopic fluid mechanics, including physical properties; hydrostatics; basic rate laws for mass, energy, and momentum; and the fundamental principles of flow through pumps, pipes, and other equipment. Part II turns to microscopic fluid mechanics, which covers

  • Differential equations of fluid mechanics
  • Viscous-flow problems, some including polymer processing
  • Laplace's equation, irrotational, and porous-media flows
  • Nearly unidirectional flows, from boundary layers to lubrication, calendering, and thin-film applications
  • Turbulent flows, showing how the k/ε method extends conventional mixing-length theory
  • Bubble motion, two-phase flow, and fluidization
  • Non-Newtonian fluids, including inelastic and viscoelastic fluids
  • Microfluidics and electrokinetic flow effects including electroosmosis, electrophoresis, streaming potentials, and electroosmotic switching
  • Computational fluid mechanics with FlowLab and COMSOL Multiphysics

Fluid Mechanics for Chemical Engineers, Second Edition, with Microfluidics and CFD, includes 83 completely worked practical examples, several of which involve FlowLab and COMSOL Multiphysics. There are also 330 end-of-chapter problems of varying complexity, including several from the University of Cambridge chemical engineering examinations. The author covers all the material needed for the fluid mechanics portion of the Professional Engineer's examination.

The author's Web site, www.engin.umich.edu/~fmche/, provides additional notes on individual chapters, problem-solving tips, errata, and more.

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Editorial Reviews

Booknews
A text for undergraduates and first-year graduate students, offering four chapters on macroscopic or relatively large-scale phenomena, followed by eight chapters on microscopic or relatively small-scale phenomena. Provides numerous real-world examples and chapter problems of increasing complexity, including several from the University of Cambridge chemical engineering exams, and covers all the material necessary to pass the fluid mechanics portion of the Professional Engineer's exam. Annotation c. by Book News, Inc., Portland, Or.
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Product Details

Meet the Author

James O. Wilkes is Professor Emeritus of Chemical Engineering at the University of Michigan, where he served as department chairman and assistant dean for admissions. From 1989 to 1992, he was an Arthur F. Thurnau Professor. Wilkes coauthored Applied Numerical Methods (Wiley, 1969) and Digital Computing and Numerical Methods (Wiley, 1973). He received his bachelor s degree from the University of Cambridge and his M.S. and Ph.D. in chemical engineering from the University of Michigan. His research interests involve numerical methods for solving a wide variety of engineering problems.

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Read an Excerpt

This text has evolved from a need for a single volume that embraces a wide range of topics in fluid mechanics. The material consists of two parts—four chapters on macroscopic or relatively large-scale phenomena, followed by ten chapters on microscopic or relatively small-scale phenomena. Throughout, I have tried to keep in mind topics of industrial importance to the chemical engineer. The scheme is summarized in the following list of chapters.
Part I—Macroscopic Fluid Mechanics
1. Introduction to Fluid Mechanics
2. Mass, Energy, and Momentum Balances
3. Fluid Friction in Pipes
4. Flow in Chemical Engineering Equipment

Part II—Microscopic Fluid Mechanics
5. Differential Equations of Fluid Mechanics
6. Solution of Viscous-Flow Problems
7. Laplace's Equation, Irrotational and Porous-Media Flows
8. Boundary-Layer and Other Nearly Unidirectional Flows
9. Turbulent Flow
10. Bubble Motion, Two-Phase Flow, and Fluidization
11. Non-Newtonian Fluids
12. Microfluidics and Electrokinetic Flow Effects
13. An Introduction to Computational Fluid Dynamics and FlowLab
14. COMSOL (FEMLAB) Multi-physics for Solving Fluid Mechanics Problems

In our experience, an undergraduate fluid mechanics course can be based on Part I plus selected parts of Part II, and a graduate course can be based on much of Part II, supplemented perhaps by additional material on topics such as approximate methods and stability.

Second edition. I have attempted to bring the book up to date by the major addition of Chapters 12, 13, and 14—one on microfluidics and two on CFD (computational fluid dynamics). The choice of software for the CFD presented a difficulty; for various reasons, I selected FlowLab and COMSOL Multiphysics, but there was no intention of "promoting" these in favor of other excellent CFD programs.1 The use of CFD examples in the classroom really makes the subject come "alive," because the previous restrictive necessities of "nice" geometries and constant physical properties, etc., can now be lifted. Chapter 9, on turbulence, has also been extensively rewritten; here again, CFD allows us to venture beyond the usual flow in a pipe or between parallel plates and to investigate further practical situations such as turbulent mixing and recirculating flows.

Example problems. There is an average of about six completely worked examples in each chapter, including several involving COMSOL (dispersed throughout Part II) and FlowLab (all in Chapter 13). The end of each example is marked by a small, hollow square. All the COMSOL examples have been run on a Macintosh G4 computer using FEMLAB 3.1, but have also been checked on a PC; those using a PC or other releases of COMSOL/FEMLAB may encounter slightly different windows than those reproduced here. The format for each COMSOL example is: (a) problem statement, (b) details of COMSOL implementation, and (c) results and discussion (however, item (b) can easily be skipped for those interested only in the results).

The numerous end-of-chapter problems have been classified roughly as easy (E), moderate (M), or difficult/lengthy (D). The University of Cambridge has given permission—kindly endorsed by Professor J.F. Davidson, F.R.S.—for several of their chemical engineering examination problems to be reproduced in original or modified form, and these have been given the additional designation of "(C)".

Further information. The website http://www.engin.umich.edu/~fmche is maintained as a "bulletin board" for giving additional information about the book—hints for problem solutions, errata, how to contact the authors, etc.—as proves desirable. My own Internet address is wilkes@umich.edu. The text was composed on a Power Macintosh G4 computer using the TEXtures "typesetting" program. Eleven-point type was used for the majority of the text. Most of the figures were constructed using MacDraw Pro, Excel, and KaleidaGraph.

Professor Terence Fox, to whom this book is dedicated, was a Cambridge engineering graduate who worked from 1933 to 1937 at Imperial Chemical Industries Ltd., Billingham, Yorkshire. Returning to Cambridge, he taught engineering from 1937 to 1946 before being selected to lead the Department of Chemical Engineering at the University of Cambridge during its formative years after the end of World War II. As a scholar and a gentleman, Fox was a shy but exceptionally brilliant person who had great insight into what was important and who quickly brought the department to a preeminent position. He succeeded in combining an industrial perspective with intellectual rigor. Fox relinquished the leadership of the department in 1959, after he had secured a permanent new building for it (carefully designed in part by himself).

Fox was instrumental in bringing Peter Danckwerts, Kenneth Denbigh, John Davidson, and others into the department. He also accepted me in 1956 as a junior faculty member, and I spent four good years in the CambridgeUniversity Department of Chemical Engineering. Danckwerts subsequently wrote an appreciation2 of Fox's talents, saying, with almost complete accuracy: "Fox instigated no research and published nothing." How times have changed—today, unless he were known personally, his résumé would probably be cast aside and he would stand little chance of being hired, let alone of receiving tenure! However, his lectures, meticulously written handouts, enthusiasm, genius, and friendship were a great inspiration to me, and I have much pleasure in acknowledging his positive impact on my career.

James O. Wilkes August 18, 2005

1. The software name "FEMLAB" was changed to "COMSOL Multiphysics" in September 2005, the first release under the new name being COMSOL 3.2.

2. P.V. Danckwerts, "Chemical engineering comes to Cambridge," The Cambridge Review, pp. 53-55, February28, 1983.

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Table of Contents

Preface.

I. MACROSCOPIC FLUID MECHANICS.

1. Introduction to Fluid Mechanics.

1.1 Fluid Mechanics in Chemical Engineering

1.2 General Concepts of a Fluid

1.3 Stresses, Pressure, Velocity, and the Basic Laws

1.4 Physical Properties - Density, Viscosity, and Surface Tension

1.5 Units and Systems of Units

Example 1.1 - Units Conversion

Example 1.2 - Mass of Air in a Room

1.6 Hydrostatics

Example 1.3 - Pressure in an Oil Storage Tank

Example 1.4 - Multiple Fluid Hydrostatics

Example 1.5 - Pressure Variations in a Gas

Example 1.6 - Hydrostatic Force on a Curved Surface

Example 1.7 - Application of Archimedes?f Law

1.7 Pressure Change Caused by Rotation

Example 1.8 - Overflow from a Spinning Container

Problems for Chapter 1

2. Mass, Energy, and Momentum Balances.

2.1 General Conservation Laws

2.2 Mass Balances

Example 2.1 - Mass Balance for Tank Evacuation

2.3 Energy Balances

Example 2.2 - Pumping n-Pentane

2.4 Bernoulli’s Equation

2.5 Applications of Bernoulli?fs Equation

Example 2.3 - Tank Filling

2.6 Momentum Balances

Example 2.4 - Impinging Jet of Water

Example 2.5 - Velocity of Wave on Water

Example 2.6 - Flow Measurement by a Rotameter

2.7 Pressure, Velocity, and Flow Rate Measurement

Problems for Chapter

3. Fluid Friction in Pipes.

3.1 Introduction

3.2 Laminar Flow

Example 3.1 - Polymer Flow in a Pipeline

3.3 Models for Shear Stress

3.4 Piping and Pumping Problems

Example 3.2 - Unloading Oil from a Tanker

Specified Flow Rate and Diameter

Example 3.3 - Unloading Oil from a Tanker

Specified Diameter and Pressure Drop

Example 3.4 - Unloading Oil from a Tanker

Specified Flow Rate and Pressure Drop

Example 3.5 - Unloading Oil from a Tanker

Miscellaneous Additional Calculations

3.5 Flow in Noncircular Ducts

Example 3.6 - Flow in an Irrigation Ditch

3.6 Compressible Gas Flow in Pipelines

3.7 Compressible Flow in Nozzles

3.8 Complex Piping Systems

Example 3.7 - Solution of a Piping/Pumping Problem

Problems for Chapter 3

4. Flow in Chemical Engineering Equipment.

4.1 Introduction

4.2 Pumps and Compressors

Example 4.1 - Pumps in Series and Parallel

4.3 Drag Force on Solid Particles in Fluids

Example 4.2 - Manufacture of Lead Shot

4.4 Flow Through Packed Beds

Example 4.3 - Pressure Drop in a Packed-Bed Reactor

4.5 Filtration

4.6 Fluidization

4.7 Dynamics of a Bubble-Cap Distillation Column

4.8 Cyclone Separators

4.9 Sedimentation

4.10 Dimensional Analysis

Example 4.4 - Thickness of the Laminar Sublayer

Problems for Chapter 4

II. MICROSCOPIC FLUID MECHANICS.

5. Differential Equations of Fluid Mechanics.

5.1 Introduction to Vector Analysis

5.2 Vector Operations

Example 5.1 - The Gradient of a Scalar

Example 5.2 - The Divergence of a Vector

Example 5.3 - An Alternative to the Differential Element

Example 5.4 - The Curl of a Vector

Example 5.5 - The Laplacian of a Scalar

5.3 Other Coordinate Systems

5.4 The Convective Derivative

5.5 Differential Mass Balance

Example 5.6 - Physical Interpretation of the Net Rate of Mass Outflow

Example 5.7 - Alternative Derivation of the Continuity Equation

5.6 Differential Momentum Balances

5.7 Newtonian Stress Components in Cartesian Coordinates

Example 5.8 - Constant-Viscosity Momentum Balances in Terms of Velocity Gradients

Example 5.9 - Vector Form of Variable-Viscosity Momentum Balance

Problems for Chapter 5

6. Solution of Viscous-Flow Problems.

6.1 Introduction

6.2 Solution of the Equations of Motion in Rectangular Coordinates

Example 6.1 - Flow Between Parallel Plates

6.3 Alternative Solution Using a Shell Balance

Example 6.2 - Shell Balance for Flow Between Parallel Plates

Example 6.3 - Film Flow on a Moving Substrate

Example 6.4 - Transient Viscous Diffusion of Momentum (FEMLAB)

6.4 Poiseuille and Couette Flows in Polymer Processing

Example 6.5 - The Single-Screw Extruder

Example 6.6 - Flow Patterns in a Screw Extruder (FEMLAB)

6.5 Solution of the Equations of Motion in Cylindrical x Table of Contents Coordinates

Example 6.7 - Flow Through an Annular Die

Example 6.8 - Spinning a Polymeric Fiber

6.6 Solution of the Equations of Motion in Spherical Coordinates

Example 6.9 - Analysis of a Cone-and-Plate Rheometer

Problems for Chapter 6

7. Laplace’s Equation, Irrotational and Porous-Media Flows.

7.1 Introduction

7.2 Rotational and Irrotational Flows

Example 7.1 - Forced and Free Vortices

7.3 Steady Two-Dimensional Irrotational Flow

7.4 Physical Interpretation of the Stream Function

7.5 Examples of Planar Irrotational Flow

Example 7.2 - Stagnation Flow

Example 7.3 - Combination of a Uniform Stream and a Line Sink (C)

Example 7.4 - Flow Patterns in a Lake (FEMLAB)

7.6 Axially Symmetric Irrotational Flow

7.7 Uniform Streams and Point Sources

7.8 Doublets and Flow Past a Sphere

7.9 Single-Phase Flow in a Porous Medium

Example 7.5 - Underground Flow of Water

7.10 Two-Phase Flow in Porous Media

7.11 Wave Motion in Deep Water

Problems for Chapter 7

8. Boundary-Layer Aand Other Nearly Unidirectional Flows.

8.1 Introduction

8.2 Simplified Treatment of Laminar Flow Past a Flat Plate

Example 8.1 - Flow in an Air Intake

8.3 Simplification of the Equations of Motion

8.4 Blasius Solution for Boundary-Layer Flow

8.5 Turbulent Boundary Layers

Example 8.2 - Laminar and Turbulent Boundary Layers Compared

8.6 Dimensional Analysis of the Boundary-Layer Problem

8.7 Boundary-Layer Separation

Example 8.3 - Boundary-Layer Flow Between Parallel Plates (FEMLAB Library)

Example 8.4 - Entrance Region for Laminar Flow Between Flat Plates

8.8 The Lubrication Approximation

Example 8.5 - Flow in a Lubricated Bearing (FEMLAB)

8.9 Polymer Processing by Calendering

Example 8.6 - Pressure Distribution in a Calendered Sheet

8.10 Thin Films and Surface Tension

Problems for Chapter 8

9. Turbulent Flow.

9.1 Introduction

Example 9.1 - Numerical Illustration of a Reynolds Stress Term

9.2 Physical Interpretation of the Reynolds Stresse

9.3 Mixing-Length Theory

9.4 Determination of Eddy Kinematic Viscosity and Mixing Length

9.5 Velocity Profiles Based on Mixing Length Theory 486

Example 9.2 - Investigation of the von K?Larm?Lan Hypothesis

9.6 The Universal Velocity Profile for Smooth Pipes

9.7 Friction Factor in Terms of Reynolds Number for Smooth Pipes

Example 9.3 - Expression for the Mean Velocity

9.8 Thickness of the Laminar Sublayer

9.9 Velocity Profiles and Friction Factor for Rough Pipe

9.10 Blasius-Type Law and the Power-Law Velocity Profile

9.11 A Correlation for the Reynolds Stresses

9.12 Computation of Turbulence by the k/? Method

Example 9.4 - Flow Through an Orifice Plate (FEMLAB)

Example 9.5 - Turbulent Jet Flow (FEMLAB)

9.13 Analogies Between Momentum and Heat Transfer

Example 9.6 - Evaluation of the Momentum/Heat-Transfer Analogies

9.14 Turbulent Jets

Problems for Chapter 9

10. Bubble Motion, Two-Phase Flow, and Fluidization.

10.1 Introduction

10.2 Rise of Bubbles in Unconfined Liquids

Example 10.1 - Rise Velocity of Single Bubbles

10.3 Pressure Drop and Void Fraction in Horizontal Pipes

Example 10.2 - Two-Phase Flow in a Horizontal Pipe

10.4 Two-Phase Flow in Vertical Pipes

Example 10.3 - Limits of Bubble Flow

Example 10.4 - Performance of a Gas-Lift Pump

Example 10.5 - Two-Phase Flow in a Vertical Pipe

10.5 Flooding

10.6 Introduction to Fluidization

10.7 Bubble Mechanics

10.8 Bubbles in Aggregatively Fluidized Beds

Example 10.6 - Fluidized Bed with Reaction (C)

Problems for Chapter 10

11. Non-Newtonian Fluids.

11.1 Introduction

11.2 Classification of Non-Newtonian Fluids

11.3 Constitutive Equations for Inelastic Viscous Fluids

Example 11.1 - Pipe Flow of a Power-Law Fluid

Example 11.2 - Pipe Flow of a Bingham Plastic

Example 11.3 - Non-Newtonian Flow in a Die (FEMLAB Library)

11.4 Constitutive Equations for Viscoelastic Fluids

11.5 Response to Oscillatory Shear

11.6 Characterization of the Rheological Properties of Fluids

Example 11.4 - Proof of the Rabinowitsch Equation

Example 11.5 - Working Equation for a Coaxial Cylinder Rheometer: Newtonian Fluid

Problems for Chapter 11

12. Microfluidics and Electrokinetic Flow Effects.

12.1 Introduction

12.2 Physics of Microscale Fluid Mechanics

12.3 Pressure-driven Flow Through Microscale Tubes

Example 12.1 - Calculation of Reynolds Numbers

12.4 Mixing, Transport, and Dispersion

12.5 Species, Energy, and Charge Transport

12.6 The Electrical Double Layer and Electrokinetic Phenomena

Example 12.2 - Relative Magnitudes of Electroosmotic and Pressure-driven Flow

Example 12.3 - Electroosmotic Flow Around a Particle

Example 12.4 - Electroosmosis in a Microchannel (FEMLAB)

Example 12.5 - Electroosmotic Switching in a Branched Microchannel (FEMLAB)

12.7 Measuring the Zeta Potential

Example 12.6 - Magnitude of Typical Streaming Potentials

12.8 Electroviscosity

12.9 Particle and Macromolecule Motion in Microfluidic Channels

Example 12.7 - Gravitational and Magnetic Settling of Assay Beads

Problems for Chapter 12

13. An Introduction to Computational Fluid Dynamics and Flowlab.

13.1 Introduction and Motivation

13.2 Numerical Methods

13.3 Learning CFD by Using FlowLab

13.4 Practical CFD Examples

Example 13.1 - Developing Flow in a Pipe Entrance Region (FlowLab)

Example 13.2 - Pipe Flow Through a Sudden Expansion (FlowLab)

Example 13.3 - A Two-Dimensional Mixing Junction (FlowLab)

Example 13.4 - Flow Over a Cylinder (FlowLab)

References for Chapter 13

14. Femlab for Solving Fluid Mechanics Problems.

14.1 Introduction to FEMLAB

14.2 How to Run FEMLAB

Example 14.1 - Flow in a Porous Medium with an Obstruction (FEMLAB)

14.3 Draw Mode

14.4 Solution and Related Modes

14.5 Fluid Mechanics Problems Solvable by FEMLAB

Problems for Chapter 14

Appendix A: Useful Mathematical Relationships.

Appendix B: Answers to the True/False Assertions.

Appendix C: Some Vector and Tensor Operations.

Index.

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Preface

This text has evolved from a need for a single volume that embraces a wide range of topics in fluid mechanics. The material consists of two parts--four chapters on macroscopic or relatively large-scale phenomena, followed by ten chapters on microscopic or relatively small-scale phenomena. Throughout, I have tried to keep in mind topics of industrial importance to the chemical engineer. The scheme is summarized in the following list of chapters.

Part I--Macroscopic Fluid Mechanics
1. Introduction to Fluid Mechanics
2. Mass, Energy, and Momentum Balances
3. Fluid Friction in Pipes
4. Flow in Chemical Engineering Equipment

Part II--Microscopic Fluid Mechanics
5. Differential Equations of Fluid Mechanics
6. Solution of Viscous-Flow Problems
7. Laplace's Equation, Irrotational and Porous-Media Flows
8. Boundary-Layer and Other Nearly Unidirectional Flows
9. Turbulent Flow
10. Bubble Motion, Two-Phase Flow, and Fluidization
11. Non-Newtonian Fluids
12. Microfluidics and Electrokinetic Flow Effects
13. An Introduction to Computational Fluid Dynamics and FlowLab
14. COMSOL (FEMLAB) Multi-physics for Solving Fluid Mechanics Problems

In our experience, an undergraduate fluid mechanics course can be based on Part I plus selected parts of Part II, and a graduate course can be based on much of Part II, supplemented perhaps by additional material on topics such as approximate methods and stability.

Second edition. I have attempted to bring the book up to date by the major addition of Chapters 12, 13, and 14--one on microfluidics and two on CFD (computational fluid dynamics). The choice of software for the CFD presented a difficulty; for various reasons, I selected FlowLab and COMSOL Multiphysics, but there was no intention of "promoting" these in favor of other excellent CFD programs. 1 The use of CFD examples in the classroom really makes the subject come "alive," because the previous restrictive necessities of "nice" geometries and constant physical properties, etc., can now be lifted. Chapter 9, on turbulence, has also been extensively rewritten; here again, CFD allows us to venture beyond the usual flow in a pipe or between parallel plates and to investigate further practical situations such as turbulent mixing and recirculating flows.

Example problems. There is an average of about six completely worked examples in each chapter, including several involving COMSOL (dispersed throughout Part II) and FlowLab (all in Chapter 13). The end of each example is marked by a small, hollow square. All the COMSOL examples have been run on a Macintosh G4 computer using FEMLAB 3.1, but have also been checked on a PC; those using a PC or other releases of COMSOL/FEMLAB may encounter slightly different windows than those reproduced here. The format for each COMSOL example is: (a) problem statement, (b) details of COMSOL implementation, and (c) results and discussion (however, item (b) can easily be skipped for those interested only in the results).

The numerous end-of-chapter problems have been classified roughly as easy (E), moderate (M), or difficult/lengthy (D). The University of Cambridge has given permission--kindly endorsed by Professor J.F. Davidson, F.R.S.--for several of their chemical engineering examination problems to be reproduced in original or modified form, and these have been given the additional designation of "(C)".

Further information. The website http://www.engin.umich.edu/~fmche is maintained as a "bulletin board" for giving additional information about the book--hints for problem solutions, errata, how to contact the authors, etc.--as proves desirable. My own Internet address is wilkes@umich.edu. The text was composed on a Power Macintosh G4 computer using the TEXtures "typesetting" program. Eleven-point type was used for the majority of the text. Most of the figures were constructed using MacDraw Pro, Excel, and KaleidaGraph.

Professor Terence Fox, to whom this book is dedicated, was a Cambridge engineering graduate who worked from 1933 to 1937 at Imperial Chemical Industries Ltd., Billingham, Yorkshire. Returning to Cambridge, he taught engineering from 1937 to 1946 before being selected to lead the Department of Chemical Engineering at the University of Cambridge during its formative years after the end of World War II. As a scholar and a gentleman, Fox was a shy but exceptionally brilliant person who had great insight into what was important and who quickly brought the department to a preeminent position. He succeeded in combining an industrial perspective with intellectual rigor. Fox relinquished the leadership of the department in 1959, after he had secured a permanent new building for it (carefully designed in part by himself).

Fox was instrumental in bringing Peter Danckwerts, Kenneth Denbigh, John Davidson, and others into the department. He also accepted me in 1956 as a junior faculty member, and I spent four good years in the CambridgeUniversity Department of Chemical Engineering. Danckwerts subsequently wrote an appreciation 2 of Fox's talents, saying, with almost complete accuracy: "Fox instigated no research and published nothing." How times have changed--today, unless he were known personally, his résumé would probably be cast aside and he would stand little chance of being hired, let alone of receiving tenure! However, his lectures, meticulously written handouts, enthusiasm, genius, and friendship were a great inspiration to me, and I have much pleasure in acknowledging his positive impact on my career.

James O. Wilkes
August 18, 2005

1. The software name "FEMLAB" was changed to "COMSOL Multiphysics" in September 2005, the first release under the new name being COMSOL 3.2.

2. P.V. Danckwerts, "Chemical engineering comes to Cambridge," The Cambridge Review, pp. 53-55, February28, 1983.

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