Flows and Chemical Reactions in Heterogeneous Mixtures

By Roger Prud'homme

This book - a sequel of previous publications 'Flows and Chemical Reactions' and 'Chemical Reactions in Flows and Homogeneous Mixtures' - is devoted to flows with chemical reactions in heterogeneous environments.  Heterogeneous media in this volume include interfaces and lines. They may be the site of radiation. Each type of flow is the subject of a chapter in this volume.

We consider first, in Chapter 1, the question of the generation of environments biphasic individuals: dusty gas, mist, bubble flow.  Chapter 2 is devoted to the study at the mesoscopic scale: particle-fluid exchange of momentum and heat with determination of the respective exchange coefficients. In Chapter 3, we establish simplified equations of macroscopic balance for mass, for the momentum and energy, in the case of particles of one size (monodisperse suspension).  Radiative phenomena are presented in Chapter 5.

• Language: English
• Publisher: Wiley
• Release date: Oct 30, 2014
• ISBN: 9781119054269

Book preview

Contents

Preface

List of Main Symbols

1 Generation of Multiphase Flows

1.1. Creation of suspensions of solid particles in a gaseous phase

1.2. Creation of suspensions of bubbles in a liquid

1.3. Creation of suspensions of drops in a gas

2 Problems at the Scale of a Particle

2.1. Force exerted by a fluid on a spherical particle

2.2. Heat exchanges

2.3. Combustion of a drop of fuel in an oxidizing environment

3 Simplified Model of a Non-reactive Flow with Particles

3.1. Variables characterizing the flow

3.2. Balance equations

3.3. Application to the linearized study of sound propagation in a non-reactive dilute suspension

3.4. Two-phase dilute flows in nozzles

4 Simplified Model of a Reactive Flow with Particles

4.1. Balance equations for a reactive fog

4.2. Application to a spray flame

5 Radiative Phenomena

5.1. Basic values and fundamental relations in radiative transfer

5.2. Application to the hypersonic flow of atmospheric re-entry

5.3. Application to the boundary layer above a flat plate with soot formation and radiative transfer

5.4. Application to combustion of aluminum-based solid propellants

Appendix: Concepts Surrounding the Hopf Bifurcation

Bibliography

Index

Title Page

First published 2014 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

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John Wiley & Sons, Inc.

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© ISTE Ltd 2014

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 Control Number: 2014950493

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN 978-1-84821-785-0

Preface

This book – a continuation of two previous publications, respectively dealing with "Flows and chemical reactions [PRU 12] and Flows and chemical reactions in homogeneous mixtures" [PRU 13] –is dedicated to "Flows and chemical reactions in heterogeneous mixtures".¹

Note that the concept of heterogeneity in fluids is entirely relative [DOR 89].² The heterogeneous media presented in this volume notably include interfaces and lines. They may be the site of radiative transfer. A separate chapter is entirely dedicated to each type of flow in this book.

Multiphase flows – whether they be gaseous flows with solid or liquid particles, liquid flows with gas or vapor bubbles, or solid/liquid particles within another phase – involve exchanges between the phases, chemical reactions, disintegration and clustering of particles, as well as the various interactions between these particles.

It is necessary to model these flows in order to discover, for example, what happens in the atmosphere, or predict the performances of a fuel injection engine or determine the risk of explosion at a grain silo.

The description of fluid–particle exchanges is a problem in itself, and requires modeling of the particle at the mesoscopic scale³, to determine, the effects of friction, or of heat- and mass exchange (e.g. the regression rate of a drop of fuel in combustion, which would help calculate the necessary length for a rocket engine). We then need to transcribe these exchange laws to the space of multi-phase mixing and write the balance equations, before going on to solve them.

A variety of methods can be used to establish the balance equations on the basis of a mixture’s properties on the molecular scale. In particular, we can cite the techniques consisting of using a probabilistic balance equation, similar to that used in Chapter 2 of Volume 2 [PRU 13], and applying it to the different quantities [BAR 73, KUE 73].

It is also possible to use Direct Numerical Simulation (DNS), or the Monte Carlo method.

To begin, in Chapter 1, we shall examine the question of the generation of specific two-phase media: dusty gases, mists, bubbly flows, etc. The production of vapor for combustion is a complex problem, with multiple steps grouped together into primary atomization and secondary atomization.

Chapter 2 is given over to examination at the mesoscopic scale: exchanges of momentum and heat between a particle and a fluid, with determination of the respective exchange coefficients. Thus, we specify the values of the phenomenological coefficients which come into play in Chapter 3, where the laws of linearized TIP⁴ are applied to multiphase flows at the macroscopic scale. Then we deal with coupled problems, such as that of the combustion of a drop of fuel (already discussed, in certain specific conditions, in Appendix 3 of Volume 1 of this series [PRU 12] and in Chapter 3 of Volume 2 [PRU 13]).

In Chapter 3, we establish simplified macroscopic balance equations for mass, momentum and energy, in the case of particles of exactly the same size (i.e. a monodisperse suspension). The forms of the constitutive relations are deduced from the principles of TIP. These equations are applied to typical problems: propagation of sound, and nozzle flow of a dilute suspension of solid particles.

Next, in Chapter 4, we examine flows with evaporating droplets, by first establishing the balance equations and then applying them to the process of combustion. Balances are first established for droplets assuming they have a uniform internal temperature; then balance equations of the two-phase mixture; and gas phase equations are deduced by subtracting those of the droplets. Finally, the balance of entropy of the mixture can be written, and phenomenological relations are deduced. We then look at the case where the internal temperature of the droplets is not uniform by using a multilayer model. An example of application is the study of the oscillatory instability of a flame spray in the presence of thermo-acoustic coupling.

Radiative phenomena are presented in Chapter 5. Such phenomena are active, in particular, in the case of production of soot resulting from the combustion of hydrocarbons, but also in problems of re-entry into planetary atmospheres and the combustion of certain solid propellants. The subject is dealt with by modeling of the radiative properties and the establishment of macroscopic balance equations. The chapter goes on to discuss a number of applications.

In the Appendix, the concept of the Hopf bifurcation is introduced.

Note that the type of equations obtained and the methods used are, of course, subject to limitations. As long as the flow regime is continuous and regular (similar to a laminar regime in a homogeneous fluid), the balance equations obtained in this book remain valid, provided the initial hypotheses made are verified (particularly that of a low volume fraction of particles). However, at high velocities or by the influence of heterogeneities in concentration, turbulence can occur, at the scale of the particle and/or that of the two-phase medium. Additionally, in the case of influence by significant external stresses such as variation of the flowrate, large-scale agglomerates may appear in the multiphase medium, as indeed may pockets of fluid. The agitation of the medium may become very great, and the equations established herein become insufficient to deal with the situation. Furthermore, this volume does not discuss transient phenomena; nor does it deal with cases where the size or concentration of particles are large, such as fluidized beds or granular media. Note that porous media are not discussed here either.

Hence, the ambition of this volume is very limited, but the few cases which are examined are fairly representative both of the complexity of the issue and of what is possible to achieve with relative ease.

Roger PRUD’HOMME

September 2014

1 Remember that the volume General Equations comprised three sections: 1. Fluid media with a single component, 2. Reactive mixtures, and 3. Interfaces and lines, and that the volume Flows and Chemical Reactions in Homogeneous Mixtures comprised: 1. Nozzle flows, 2. Chemical reactors, and 3. Laminar and turbulent flames.

2 In the publication cited here, the following remarks are made about the concept of heterogeneity:

– a fluid is said to be heterogeneous if it is possible to distinguish one particle from its neighbor and we choose to make that distinction;

– the heterogeneity may refer to various aspects: differences in phase, chemical species, velocity, temperature, etc.;

– the size of the particles in question and the diagnostic tools at our disposal are two crucial elements in shaping our ability to discern these differences.

3 That is to say, intermediary between the microscopic and macroscopic scales.

4 TIP: Thermodynamics of Irreversible Processes [GRO 69a].

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