Flows and Chemical Reactions

Table of Contents

Preface

List of the Main Symbols

Chapter 1: Simple Fluids

1.1. Introduction

1.2. Key elements in deformation theory – Lagrangian coordinates and Eulerian coordinates

1.3. Key elements in thermodynamics. Reversibility, irreversible processes: viscosity, heat conduction

1.4. Balance equations in fluid mechanics. Application to incompressible and compressible perfect fluids and viscous fluids

1.5. Examples of problems with 2D and 3D incompressible perfect fluids

1.6. Examples of problems with a compressible perfect fluid: shockwave, flow in a nozzle, and characteristics theory

1.7. Examples of problems with viscous fluids

1.8. Exercises

1.9. Solutions to the exercises

Chapter 2: Reactive Mixtures

2.1. Introduction

2.2. Equations of state

2.3. Balance equations of flows of reactive mixtures

2.4. Phenomena of transfer and chemical kinetics

2.5. Couplings

Chapter 3: Interfaces and Lines

3.1. Introduction

3.2. Interfacial phenomena

3.3. Solid and fluid curvilinear media: pipes, fluid lines and filaments

3.4. Exercises

3.5. Solutions to the exercises

APPENDICES

Appendix 1: Tensors, Curvilinear Coordinates, Geometryand Kinematics of Interfaces and Lines

A1.1. Tensor notations

A1.2. Orthogonal curvilinear coordinates

A1.3. Interfacial layers

A1.4. Curvilinear zones

A1.5. Kinematics in orthogonal curvilinear coordinates

Appendix 2: Additional Aspects of Thermostatics

A2.1. Laws of state for real fluids with a single constituent

A2.2. Mixtures of real fluids

Appendix 3: Tables for Calculating Flows of Ideal Gas ( ie283_01.gif =1 .4)

A3.1. Calculating the parameters in continuous steady flow (section 1.6.6.2)

A3.2. Formulae for steady normal shockwaves

Appendix 4: Extended Irreversible Thermodynamics

A4.1. Heat balance equations in a non-deformable medium in EIT

A4.2. Application to a 1D case of heat transfer

A4.3. Application to heat transfer with the evaporation of a droplet

A4.4. Application to thermal shock

A4.5. Outline of EIT

A4.6. Applications and perspectives of EIT

Appendix 5: Rational Thermodynamics

A5.1. Introduction

A5.2. Fundamental hypotheses and axioms

A5.3. Constitutive laws

A5.4. Case of the reactive mixture

A5.5. Critical remarks

Appendix 6: Torsors and Distributors in Solid Mechanics

A6.1. Introduction

A6.2. Derivatives of torsors and distributors which depend on a single position parameter

A6.3. Derivatives of torsors and distributors dependent on two positional parameters

Appendix 7: Virtual Powers in a Medium with a Single Constituent

A7.1. Introduction

A7.2. Virtual powers of a system of n material points

A7.3. Virtual power law

A7.4. The rigid body and systems of rigid bodies

A7.5. 3D deformable continuous medium

A7.6. 1D continuous deformable medium

A7.7. 2D deformable continuous medium

Bibliography

Index

First published 2012 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

© ISTE Ltd 2012

The rights of Roger Prud’homme to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Cataloging-in-Publication Data

Prud'homme, Roger.

Flows and chemical reactions / Roger Prud'homme.

p. cm.

Includes bibliographical references and index.

ISBN 978-1-84821-425-5

1.Chemical reactions. 2. Fluid mechanics. I. Title.

QD501.P8285 2012

541'.39--dc23

2012025722

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library ISBN: 978-1-84821-425-5

Preface

Over the past several decades, numerous publications have been devoted to the subject of flows with chemical reactions. Having initially aroused the interest of combustion specialists, their behaviors have been of direct concern for scientists in the fields of process engineering, astronautics, the atmospheric and aquatic environment, and many others.

The interactions between fluid flow, heat exchange and chemical reactions are such that in numerous applications it is impossible to deal with these aspects separately.

Indeed, it is very difficult to consider, for example, combustion in a rocket engine as the superimposition of a non-reactive fluid flow, determined first, with the chemical reactions and heat exchanges to be added. The same is true in many other fields which involve highly energetic reactions in a fluid medium which is usually compressible and dilatable.

These considerations linked to applications have given birth to a new scientific discipline – Aerothermochemistry – which is primarily concerned with establishing coherent systems of equations, with their boundary and interface conditions, which enable us to model and deal with complex situations involving multiple parameters.

This book, entitled Flows and Chemical Reactions, is divided into three chapters and seven appendices, which aim to present the equations of homogeneous laminar flows, interfaces and lines.

Chapter 1 is devoted to simple fluids – that is fluids with only one chemical component. The aim is to give a concise and academic presentation of the essential principles of fluid mechanics for incompressible and compressible, non-dissipative or dissipative fluids. This chapter’s contents formed the basis for a Masters-level module taught three times at the University of Lomé between 2004 and 2011. It offers a great many exercises and their solutions.

Chapter 2 relates to the flow of fluid mixtures in the presence of chemical reactions. This chapter has been the object of numerous teaching sessions in the DEA (Masters) in the Physics of Liquids and in the Ecoles d’Ingénieurs (specialized engineering schools – Ecole des Mines, Ecole Polytechnique).

Chapter 3 is concerned with interfaces and lines. Interfaces have been discussed in numerous books and other publications over the past fifty years.

Less is known about lines, or curvilinear media. A few basic exercises are offered, along with their solutions.

The Appendices are comprised of Appendix 1 to Appendix 7; each of which contains additional information from the fields of mathematics, thermodynamics and mechanics.

Our goal in writing this book is to provide students with a reference tool in the field of fluid mechanics, which strays somewhat from the beaten track in terms of content or form. However, this book is also aimed at researchers, lecturer/researchers, and at industrial engineers in sectors concerned with reactive fluid flows.

Part of this work is made up of texts which hitherto were only available in course notes or internal reports at ONERA (French Aerospace Lab) or CNRS (National Center for Scientific Research). These texts have been reorganized and often rethought and enriched. Other elements have been reworked from the author’s recent publications. Finally, other parts result from the study of a body of literature on the subject of flows and chemical reactions.

Roger PRUD’HOMME

August 2012

List of the Main Symbols

Latin characters

Greek symbols

Subscripts, superscripts and other symbols

Chapter 1

Simple Fluids

1.1. Introduction

This first chapter is devoted to fluids with a single chemical component – that is, it may be considered an introduction to conventional fluid mechanics. However, this chapter alone cannot take the place of the numerous detailed works which deal with all, or part, of this topic.

The first two sections (1.2 and 1.3), are given over to the fundamental notions – deformation theory, Lagrangian and Eulerian coordinates, and the laws of thermodynamics – meaning that in section 1.4, we have the necessary groundwork in place to lay out the fundamental equations of fluid mechanics.

The last three sections are dedicated to the applications and solutions of the fundamental equations. Sections 1.5 and 1.6 relate to perfect¹ fluids, i.e. fluids in a state of reversible evolution in the thermodynamic sense: incompressible perfect fluids in section 1.5, and compressible perfect fluids in section 1.6. In section 1.7, we shall turn to fluids in irreversible evolution, which exhibit viscosity and heat conduction, limiting our discussion to linearized irreversible phenomena.

Each section is illustrated with examples which not only enable the reader to familiarize themselves with the notions and theorems under discussion, but which also set out some solution methods which are conventional in fluid mechanics.

In terms of the applications given, we have limited the discussion to so-called laminar flows, i.e. whose Reynolds numbers are sufficiently low for random fluctuations in values not to occur. The balance equations established herein remain valid for turbulent flows, but in that case, in order to solve them, we need either to: apply direct numerical methods, which are costly in terms of both time and resources; establish averaged equations and new constitutive relations governing the averages of fluctuation products; or, finally, use hybrid methods such as the large eddy method.

The lessons and exercises presented here were used in the teaching of a Masters-level course at the University of Lomé in 2007. They draw heavily on the pedagogical works of Germain [GER 62, GER 73, GER 87], the course taught by Carrière at ENSTA [CAR 79], the book by Guyon, Hulin and Petit [GUY 91], and those by Barrère [BAR 60, BAR 73], to cite only those people alongside whom the author has conducted research and taught. The reader could benefit from consulting the numerous publications listed in the bibliography, such as [LAN 71, ROC 67, GUI 87] and many more.

1.2. Key elements in deformation theory – Lagrangian coordinates and Eulerian coordinates

1.2.1. Strain rates

1.2.1.1. Motion of a continuous medium

Consider the motions of two neighboring points (particles), M and M', in a continuous medium characterized by the vectors ie02_01.gif and ie02_02.gif between the times t and t + dt (see Figure 1.1). Limiting ourselves to first-order terms of the expansions in a Taylor series, we have:

[1.1] 

where:

hence:

[1.2]  eqn1.2.gif

Figure 1.1. Motion in a continuous medium. Two neighboring points M and M' represented by the vectors ie03_01.gif and ie02_02.gif at time t. We are interested in the evolution over time – at the next instant (t + dt) – of the vector ie03_03.gif when the field of velocities ie03_04.gif in the vicinity of the point M is known

As we shall see, the tensor ie03_05.gif contains the local, instantaneous strain rates of the continuum or continuous medium (be it solid or fluid).

1.2.1.2. Rate of dilatation

The infinitesimal quantity ie03_06.gif , the volume of an elementary moving domain, where: ie03_07.gif , enables us to express the rate of dilatation in terms of volume:

[1.3]  eqn1.3.gif

1.2.1.2.1. Demonstration

We write:

For equ04-01a.gif , we find:

1.2.1.3. Conservation of mass

The mass contained in ie04_01.gif is preserved during the motion. If rho.gif is the local density, we have: ie04_02.gif , which is constant throughout the motion. Hence ie04_03.gif , so:

[1.4]  eqn1.4.gif

This relation is, of course, compatible with equation [1.3].

1.2.1.4. Tensors which characterize the strain rates

ie04_04.gif is the tensor velocity gradient and ie04_05.gif is the corresponding matrix.

We can write:

where ie04_06.gif is the matrix of the strain rates, and ie04_07.gif is the matrix of the rates of rotation. The matrix ie04_08.gif is symmetric, and the matrix ie04_09.gif is antisymmetric.

If ie05_01.gif is a material vector (matrix: ie05_02.gif ), then its time derivative is ie05_03.gif . We obtain ie05_04.gif because, if the subscripts 1, 2 and 3 denote the coordinates:

[1.5] 

For the motion of rigid bodies ie05_05.gif and ie05_06.gif in any non-deformable solid (S).

1.2.1.5. Acceleration

[1.6]  eqn1.6.gif

1.2.1.6. Material derivatives of integrals

1.2.1.6.1. Material derivative of the volume integral of a derivable quantity C

By applying the Stokes-Ostrogradsky theorem, we obtain:

[1.7] 

where nu.gif is a closed moving volume with boundary ie05_07.gif ², and the external normal to the surface ie05_08.gif is denoted by ie05_09.gif . In effect:

1.2.1.6.2. Material derivative of the volume integral of a quantity C derivable by parts

[1.8]  eqn1.8.gif

where ie06_01.gif is the jump experienced by ie06_02.gif on crossing the surface of discontinuity (Σ) in the direction (−) to (+) (see Figure 1.2).

1.2.1.6.3. Material derivative of a