Table Of ContentProcess Modelling and
Simulation with
Finite Element Methods
SERIES ON STABILITY, VIBRATION AND CONTROL OF SYSTEMS
Founder and Editor: Ardeshir Guran
Co-Editors: M. Cloud 81 W. B. Zimmerman
About the Series
Rapid developments in system dynamics and control, areas related to many other
topics in applied mathematics, call for comprehensive presentations of current
topics. This series contains textbooks, monographs, treatises, conference proceed-
ings and a collection of thematically organized research or pedagogical articles
addressing dynamical systems and control.
The material is ideal for a general scientific and engineering readership, and is
also mathematically precise enough to be a useful reference for research specialists
in mechanics and control, nonlinear dynamics, and in applied mathematics and
physics.
Selected Volumes in Series A
VOl. 3 Vibration Analysis of Plates by the Superposition Method
Author: Daniel J. Gorman
VOI. 4 Asymptotic Methods in Buckling Theory of Elastic Shells
Authors: P. E. Tovstik and A. L. Smirinov
VOl. 5 Generalized Point Models in Structural Mechanics
Author: 1. V. Andronov
Vol. 6 Mathematical Problems of Control Theory: An Introduction
Author: G. A. Leonov
VOl. 7 Analytical and Numerical Methods for Wave Propagation in
Fluid Media
Author: K. Murawski
VOl. 8 Wave Processes in Solids with Microstructure
Author: V. 1. Erofeyev
VOl. 9 Amplification of Nonlinear Strain Waves in Solids
Author: A. V. Porubov
Vol. 10 Spatial Control of Vibration: Theory and Experiments
Authors: S. 0. Reza Moheimani, D. Halim, and A. J. Fleming
Vol. 11 Selected Topics in Vibrational Mechanics
Editor: 1. Blekhman
Vol. 12 The Calculus of Variations and Functional Analysis: With Optimal Control
and Applications in Mechanics
Authors: L. P. Lebedev and M. J. Cloud
Vol. 13 Multiparameter Stability Theory with Mechanical Applications
Authors: A. P. Seyranian and A. A. Mailybaev
Vol. 14 Stability of Stationary Sets in Control Systems with Discontinuous
Nonlinearities
Authors: V. A. Yakubovich, G. A. Leonov and A. Kh. Gelig
SERIES ON STABILITY, VIBRATION AND CONTROL OF SYSTEMS
Series A Volume 15
~~
Founder & Editor: Ardkshir Guran
Co-Editors: M. Cloud & W. 6. Zimmerman
Process Modelling and
Simulation with
Finite Element Methods
William B. J. Zimmerman
University of Sheffield, UK
K
World Scientific
-
NEW JERSEY * LONDON SINGAPORE * BElJlNG SHANGHAI * HONG KONG TAIPEI * CHENNAI
Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
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British Library Cataloguing-in-PublicationD ata
A catalogue record for this book is available from the British Library.
PROCESS MODELLING AND SIMULATION WITH FINITE ELEMENT METHODS
Copyright 0 2004 by World Scientific Publishing Co. Re. Ltd.
All rights reserved. This book, or parts thereoj may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright
Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to
photocopy is not required from the publisher.
ISBN 981-238-793-5
Printed in Singapore by World ScientificP rinters (S)P te Ltd
ABOUT THE AUTHOR
Dr William B. J. Zimmerman is a Reader in Chemical and Process Engineering.
His research interests are in fluid dynamics and reaction engineering. He has
previously created modules entitled Chemical Engineering Problem Solving with
Mathematica, Modelling and Simulation in Chemical Processes, Numerical
Analysis in Chemical Engineering, and FORTRAN programming. He has been
modelling with finite element methods since 1986. He has authored over sixty
scientific and scholarly works. He is a graduate of Princeton and Stanford
Universities in Chemical Engineering, past Director of the M Sc in
Environmental and Energy Engineering, originator of the M Sc in Process Fluid
Dynamics, and a winner in US and UK national competitions of four prestigious
fellowships:
2000-5 EPSRC Advanced Research Fellow
1994-99 Royal Academy of Engineering, Zeneca Young Academic Fellow.
1991-93 NATO postdoctoral fellow in science and engineering.
1988-9 1 National Science Foundation Research Fellow
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FOREWORD
I would especially like to thank the Engineering and Physical Sciences Research
Council of the United Kingdom for the award of an Advanced Research
Fellowship on the topic of “models of helical mixing and reaction: a new
approach to chemical reaction engineering.” Without the flexibility of the
fellowship, I doubt I would have felt adventuresome enough to afford the time to
run so far - developing an intensive training module and writing a textbook on a
topic that is relatively new to me and was not envisaged when I wrote the
original research strategy for the fellowship in 1998. It has turned out to be
integral to my plans for turbulence modeling, although this is not reflected in
the book.
Johan Sundqvist and Ed Fontes of COMSOL have been most supportive of
my two projects, lending considerable resources to helping me to iron out
difficulties in modeling, contributing to the intensive modules, and providing
critiques of the draft chapters. I would have thought the crew at COMSOL
would have tired with the endless e-mails from my research group. With most
packages I use, I have no intention of being “cutting edge,” so the FAQs on the
web site usually already have my queries asked and answered. It is both novel
-
and refreshing to have identified (and sometimes solved or worked around) new
bugs or puzzles. To Ed and Johan, a wholehearted thanks for welcoming me to
the FEMLAB developers community.
Many thanks to the team of collaborators and chapter co-authors who have
encouraged this effort. Buddhi, Alex, Kiran, Jordan, Peter, George and Julia
have always had a kind word and a willingness to brainstorm and contribute.
Finally, thanks to attendees at my intensive modules for spotting
inconsistencies, patiently wading through “experimental” teaching material
(guinea pigs who are so intelligent are a rare find!), and putting up with my
sometimes convoluted explanations. Not to mention those awkward times as we
uncovered clangers in the demonstrations. It has always been a conceit of mine
that computer demonstrations should be realistic bugs and all since
- -
debugging is an integral programming skill that relies on intuition and
experience. So thanks for sharing the experience!
vii
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CONTENTS
About the Author
V
Foreword vii
Introduction to FEMLAB 1
W. B. J. Zimmerman
0.1 Overview of the Book 1
0.2 An Example from the Model Library 11
0.2.1 k-&Modelo f a Turbulent Static Mixer 12
0.2.2 Why the Tour of k-E Model of a Turbulent Static
Mixer? 14
0.3 Chapter Synopsis 16
References 22
1 FEMLAB and the Basics of Numerical Analysis 23
W. B. J. Zimmerman
1.1 Introduction 23
1.2 Method 1: Root Finding 24
1.2.1 Root Finding: A Simple Application of the FEMLAB
Nonlinear Solver 25
1.2.2 Root Finding: Application to Flash Distillation 29
1.3 Method 2: Numerical Integration by Marching 33
1.3.1 Numerical Integration: A Simple Example 35
1.3.2 Numerical Integration: Tubular Reactor Design 38
1.4 Method 3: Numerical Integration of Ordinary Differential
Equations 44
1.5 Method 4: Linear Systems Analysis 49
1.5.1 Heat Transfer in a Nonuniform Medium 55
1.6 Summary 59
References 59
2 Partial Differential Equations and the Finite Element Method 63
W. B. J. Zimmerman and B. N. Hewakandamby
2.1 Introduction 63
2.1.1 Poisson’s Equation: An Elliptic PDE 66
2.1.2 The Diffusion Equation: A Parabolic PDE 70
2.1.3 The Wave Equation: A Hyperbolic PDE 75
ix
