Turbulent Multiphase Flows with Heat and Mass Transfer E-Book

Roland Borghi

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 729

Veröffentlichungsjahr: 2013

Beschreibung

Numerous industrial systems or natural environments involve multiphase flows with heat and mass transfer. The authors of this book present the physical modeling of these flows, in a unified way, which can include various physical aspects and several levels of complexity.

Thermal engineering and nuclear reactors; the extraction and transport of petroleum products; diesel and rocket engines; chemical engineering reactors and fluidized beds; smoke or aerosol dispersion; landslides and avalanches − the modeling of multiphase flows with heat and mass transfer for all these situations can be developed following a common methodology. This book is devoted to the description of the mathematical bases of how to incorporate adequate physical ingredients in agreement with known experimental facts and how to make the model evolve according to the required complexity.

Leseprobe

Table of Contents

Acknowledgments

Introduction

PART 1: Approach and General Equations

Chapter 1: Towards a Unified Description of Multiphase Flows

1.1. Continuous approach and kinetic approach

1.2. Eulerian–Lagrangian and Eulerian formulations

Chapter 2: Instant Equations for a Piecewise Continuous Medium

2.1. Integral and differential forms of balance equations

2.2. Phase mass balance equations in a piecewise continuous medium

2.3. Momentum balances

2.4. Energy balances

2.5. Position and interface area balance equations

2.6. Extension for a fluid phase that is a mixture

2.7. Completing the description of the medium

Chapter 3: Description of a “Mean Multiphase Medium”

3.1. The need for a mean description

3.2. How are mean values defined?

3.3. Which average to choose, according to their advantages and disadvantages?

Chapter 4: Equations for the Mean Continuous Medium

4.1. Global balance equations for the mean medium

4.2. Balance equations for the phases of a mean medium

4.3. Complete representation of the mean medium

4.4. Mean equations of state

4.5. Extensions

4.6. Boundary conditions

PART 2: Modeling: A Single Approach Adaptable to Multiple Applications

Chapter 5: The Modeling of Interphase Exchanges

5.1. General methodology

5.2. Interface between phases and its mean area per unit of volume

5.3. Forces of contact and friction between phases

5.4. Heat transfers at the surface of a particle, without mass exchange

5.5. Heat and mass transfers during boiling

5.6. Mass and heat exchanges by vaporization

Chapter 6: Modeling Turbulent Dispersion Fluxes

6.1. Global modeling

6.2. “Multifluid” modeling

Chapter 7: Modeling the Mean Gas–Liquid Interface Area per Unit Volume

7.1. Introduction

7.2. Initial equation for the mean interface area per unit volume

7.3. Model of the mean interface area during the “atomization” of a liquid jet

7.4. Effects of vaporization on the interface area

Chapter 8: “Large Eddy Simulation” Style Models

8.1. Introduction

8.2. Filtered equations and the nature of the models to be provided

8.3. Classic LES modeling for SGS additional fluxes

8.4. Subgrid modeling of the interface area per unit volume

8.5. Partially Integrated Turbulence Modeling

Chapter 9: Contribution of Thermodynamics of Irreversible Processes

9.1. Global two-phase medium models

9.2. Contribution of thermodynamics to multifluid models

Chapter 10: Experimental Methods

10.1. Introduction

10.2. Intrusive methods

10.3. Non-intrusive methods

10.4. Advanced optical methods

Chapter 11: Some Experimental Results Pertaining to Multiphase Flow Properties that Are Still Little Understood

11.1. Atomization/fragmentation of liquid jets

11.2. Isolated bubbles, bubbles in swarm and their effects on carrier fluid

11.3. Boiling crisis

PART 3: From Fluidized Beds to Granular Media

Chapter 12: Fluidized Beds

12.1. Introduction

12.2. Complete models for the dynamics of fluidized beds

12.3. Global models for chemical conversion in fluidized beds

12.4. Global models for heat transfers in fluidized beds

12.5. Conclusion

Chapter 13: Generalizations for Granular Media

13.1. Introduction

13.2. Balance equations for mean granular media

13.3. Necessary closure approximations

13.4. Some already proposed methods

Chapter 14: Modeling of Cauchy Tensor of Sliding Contacts

14.1. Hypotheses and basic equations

14.2. Unclosed balance equation for Cauchy tensor of sliding contact

14.3. Closure approximations for irreversible terms

Chapter 15: Modeling the Kinetic Cauchy Stress Tensor

15.1. Prandtl–Bagnold modeling

15.2. K-lt or “turbulent granular gas” modeling

15.3. Toward a general model for all regimes

15.4. Boundary conditions at walls

PART 4: Studying Fluctuations and Probability Densities

Chapter 16: Fluctuations of the Gas Phase in Reactive Two-Phase Media

16.1. Specificities of reactive two-phase media

16.2. Probability density of composition fluctuations of the gas phase

16.3. Modeling the terms due to exchanges between phases

16.4. Modeling micromixing and turbulent dispersion

16.5. Practical use of PDF equations

Chapter 17: Temperature Fluctuations in Condensed Phases

17.1. Problems

17.2. Instantaneous equation for the temperature of the liquid phase

17.3. Equation for the PDF of the temperature of the liquid

17.4. Closure of the equation of the temperature PDF

Chapter 18: Study of the PDF for Velocity Fluctuations and Sizes of Parcels

18.1. Phase velocity PDF equation

18.2. Modeling the exchanges between phases and the internal interactions

18.3. Practical calculation of PDF

18.4. The study of the sizes of the dispersed phase parcels

18.5. Eulerian–Lagrangian simulation of dispersed media

Bibliography

Index

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:

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The rights of Roland Borghi and Fabien Anselmet to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2013951424

British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN: 978-1-84821-617-4

Acknowledgments

The idea for this book emerged from several thesis projects at the University of Rouen where Roland Borghi has been working for more than fifteen years. Its writing owes much to our many interactions and discussions with doctoral candidates at the university as well as with our colleagues at the CORIA laboratory.

We must also make special mention of our many unfailingly open and stimulating discussions with Olivier Simonin, over a period of more than 30 years, and in the process of completing this book. Likewise, the arduous study of probability densities was greatly helped by years of exchange with Vladimir Sabelnikov, who even volunteered to proofread and critique the corresponding part of this book. Nevertheless, as usual, any error left in the text would only be our responsibility. We extend our gratitude to one and all, especially Olivier and Vladimir.

Fabien Anselmet offers warm personal thanks to a few colleagues who are his elders, including Roland Borghi of this book, for having introduced him to the fascinating world of turbulence, multiphase flows, drops and droplets. This book can consequently be considered as the result of thirty years of collaboration and discussion.

Introduction

I.1. The significance of multiphase flows and their modeling

Many industrial systems bring into play, in one way or the other, multiphase media involving the combination of liquids and gases, non-miscible liquids, fluids and solids.

Nuclear reactors (whether they use boiling or pressurized water) possess a cooling circuit where, in certain parts of this circuit, a mixture of water and vapor circulates, with water vapor forming on contact with hot walls needing to be cooled and drops of liquid water forming on contact with cold walls needing to be heated. Numerous other thermal engineering facilities possess this type of circuit for transferring heat, in order either to use this energy elsewhere or simply to prevent the machinery from being destroyed by the heat.

The extraction and transportation of oil products is done using conduits within which media are flowing with two or more phases: liquids of different densities and viscosities, gases and even solids. Problems of icing in aeronautics (on the leading edges of wings or ailerons or in Pitot tubes, etc.) also necessitate studying a humid air medium with drops of water flowing in the immediate vicinity of the wall. The short-distance transport of pulverulent materials such as wheat, sawdust and grain is done by blowing air loaded with these solid particles through ducts.

In liquid-fuel rocket engines used in space launchers, as well as in diesel engines, the combustion chamber contains a mixture of vaporizing droplets and combusting gases that give off a considerable release of heat in an astonishingly small volume. A combustible or oxidant liquid, or a mixture of both, is injected in tiny droplets into the combustion chamber, where these drops vaporize and the vapors can burn together, in a steady regime in a rocket engine and in a periodic regime in cycles of a diesel engine.

Fuel burners in glassworks furnaces, or vapor generators in thermal power stations, also inject jets of droplets of fuel into the zone of reacting gases. They produce not only heat and burned gases, but also smoke in which very small particles of carbon are dispersed, and the control of these particles is critical; they allow high heat transfers via radiation in furnaces, but can lead to significant air pollution from chimney exhaust.

Chemical engineering uses several types of gas–liquid reactors at controlled temperature, which are intended to produce specific chemical products rather than heat. Liquid and gaseous reactants are mixed as effectively as possible in order to be able to allow various chemical reactions at the interface between phases. Many chemical reactors also use a catalyst, which is most often in the form of a solid dispersed phase, and these reactors therefore bring multiphase flows into play.

Fluidized beds are currently the most effective devices for burning coal: air is blown forcefully through a highly dense dispersed solid phase composed of particles of dolomite and small particles of coal, enabling exchanges of energy among these three phases, which then causes and maintains chemical reactions. The energy released by combustion produces water vapor by the intermediary of a heat exchanger, the tubes of which can be closer to the combustion zone. The system not only enables adequate homogeneity of the temperature field but also maintains this temperature at around 1,300 K by a voiding the overproduction of NO, while stabilizing combustion at the same time. In addition, dolomite absorbs sulfur and reduces SO2 emissions. There are also recirculating fluidized beds in which solid phases are entrained by the gas phase, recovered at the exit and reinjected at the entry to perfect combustion, even in the highest mass flow rate conditions. These are also easily transposable for the combustion of different types of combustibles, ranging from gas mixtures to various types of waste. Fluidized beds are also used in chemical engineering, or simply to dry solid particles, or to manufacture various types of powders.

Multiphase media often play in nature as well. Clouds contain tiny droplets of (non-pure) water along with particles of ice or snowflakes; agriculture utilizes jets of drops or droplets for the watering or treatment of plants. The dispersion of smoke or other natural or artificial aerosols into the atmosphere or confined gaseous environments, and the possible deposit of the solid or liquid particles they contain, is a source of ongoing problems that are difficult to solve or control. Landslides, avalanches, and flows of sand or various types of debris are also natural examples of flows of multiphase media, the behavior of which is difficult to predict. Soil, even stable soil, is always a multiphase medium containing at least one liquid phase, which is usually water, but air for backfill and ballast for railroads, and in civil engineering we must always have control over the strains of these media that is as complete as possible.

In all of these industrial or natural situations, the overall medium behaves very often like a fluid. This is obviously the case when there is no solid phase, but as soon as one fluid phase exists the medium can be in flow, at least in some parts. This makes it desirable to be able to study these media in the same way as classic fluids, for which we have nearly 100 years of accumulated knowledge and experience. Moreover, these flows almost always display a highly random nature, both in the positions of phases and because they show that velocity fluctuations develop, quite often, similarly to turbulent flows. Experience in modeling turbulent flows should, therefore, prove extremely useful. This experience is not confined solely to questions of motion; we can also represent phenomena in fluids such as heat exchange, diffusion and mixing of various constituents, and chemical reactions, all out of equilibrium. The short descriptions provided above for multiphase situations make it clear that they also include all these phenomena, and that these phenomena give them important specificities. Therefore, it is very useful to generalize the approach used for out of equilibrium continuous fluid media to these multiphase flows.

Since the early years of the use of multiphase devices in various fields of application, a notable body of results and knowledge of an empirical or even occasionally theoretical nature has been accumulated; this has been used to develop simplified and practical approaches for the study and prediction of their characteristics. The design of industrial machinery and the interpretation and control of natural multiphase phenomena have been aided by the various separate branches of this knowledge set, each of them directly linked to a particular application. However, for the last 20 years, the desire for increasingly detailed predictions has resulted in the increasingly frequent use of the methods of continuous fluid mechanics, thus showing a certain community of approach in the different cases. Even without necessarily seeking to build theoretical representations, when we wish to understand how the various basic physical phenomena that occur on a small scale combine, we find a strong similarity of description in numerous different multiphase situations. With regard to the specific aspect of mixing, this is well emphasized in [GUY 97].

It now seems possible and interesting to attempt a unified explanation of the basic theoretical concepts used in the modeling of all multiphase media of these various applications, even though the particularities of the various different situations must explicitly be involved at a certain point. This is the aim of this book. Seeking such a unified methodology has a purpose, even a threefold purpose: first, it may provide a more complete physical understanding of each situation by bringing together information and analyses of situations that are different but using similar phenomena. Second, it may serve as a motivation to use successfully certain modeling or study tools from one field of application in another. Finally, it may render a new field of application accessible that might seem too complex at first glance. However, it is not our objective here to develop the approach to the extent of completely addressing the issues posed by the various applications, even simply those referred to earlier. First, there are too many possible applications, but above all, it would be necessary at times to enter into more details of the modeling, and at a level that would lack interest for the non-specialist. In addition, many particular aspects of certain problems are not yet sufficiently known, and they still require critical discussions, a fact that further pushes away discussions too specific for our global purposes. We wish simply to show how a unified approach can establish a common basis of representation for all of these situations, how questions of modeling emerge, which aspects are general and which are more specific to different applications, etc. To answer these questions, the first result of this unified approach will be to make certain suggestions that are the outcome of comparisons with the issues of another type of multiphase medium.

We cited granular media and even soils above as examples of multiphase media containing solid phases. Landslides and avalanches of debris are obviously multiphase flows. Soil is generally considered as a solid medium, but it is particularly interesting to study the threshold beyond which this medium flows, completely or, most often, in part, which also poses the problem of determining the expansion of the zone which is flowing. We will show how the unified method presented here contributes a new and useful point of view for granular media. When this medium is in flow, it is easy to see how it might be seen as a particular turbulent multiphase flow. Just before the threshold of flow, the medium shifts and deforms; qualifying it as a turbulent medium is not absolutely appropriate, but the motions are sudden and random (as is the initial structure itself), even if they are of limited amplitude. However, it would be interesting to push the approach toward this limit. We will not be examining every situation here in which the medium is a deformable solid, or those in which one or more liquid or gas phases flow across a porous solid. The fields of soil mechanics and porous media mechanics are well established. It would be interesting to show how certain aspects of these studies are close to the topics we will discuss here, but space does not permit it.

I.2. Modeling and its related issues

The perspective that enables a common theoretical approach begins, of course, with the acknowledgment that multiphase media are similar to continuous media when they are viewed from far enough away that their detailed structure is no longer discernable. These detailed structures, moreover, are not the objective of studies; we are dealing simply with the phenomena that we witness and undergo at our level. However, the details of these structures have consequences for these phenomena. In these conditions, we must seek to represent the media and phenomena that occur by means of a certain number of characteristic macroscopic, or averaged, variables, and pinpoint the various types of laws they must satisfy. The first task is to adequately define the way to establish these averages. These averaged variables, like all expansive physical variables, satisfy balance equations that must then be written. The detailed structures of the medium and the small-scale phenomena that enter into play within them must be taken into account, or must be modeled, using certain adapted constitutive laws. These laws are usually found with local formulations, but this will not always be the case here; it will be necessary to create new partial differential equations, and in any case they will have to be validated in a fairly general manner through experimentation. In some cases, we will see the usefulness of suggesting several equations for the same medium but that will be valid in more or less general conditions.

Of course, the situation is more complicated for a multiphase medium than for a fluid medium composed of identical molecules. First, it is necessary to be able to monitor within the medium the way in which the different phases move, i.e. to understand the composition of the medium in different places. This is similar to a situation in which the fluid is composed of different molecules. However, these different phases are also liable to have different velocities, temperature and specific agitation energies, and these variables must be clearly defined in an appropriate manner. It will, therefore, be necessary to characterize the medium, even if it is only at a macroscopic level, by more characteristic variables, some of these being new types; for example, we will see the broad utility of the mean interface area per unit of volume. It should be remembered, however, that even if we use more variables, modeling will be only an approximation; the variables we are considering will never contain the whole complexity of the media, but only the aspects that we consider are the most important, and it will be necessary in any case to model complex behaviors that remain hidden.

The methodology of modeling is, of course, similar to that of classic fluid media, which covers the detailed movements of molecules. To avoid straying too far from this example, it is helpful to consider that the characteristic scale of the multiphase structure of the medium is much smaller than the macroscopic scale in which we are interested, that is of the device containing it, as is the case for classic flows. This is the hypothesis effectively advanced for the first attempts at modeling, and it is indeed generally the case when the medium is composed of solid particles widely diluted within a continuous fluid phase. But when we are examining the fragmentation or atomization of a liquid jet into drops in a gaseous flow, the liquid core and the first ligaments or parcels of liquid are of a size comparable to the conduits within which they are moving, while the drops being produced are expected to be much smaller. For flows in which bubbles of various sizes are transported within a liquid, or flows of an emulsion of two non-miscible liquids, there are conditions in which the bubbles, drops or inclusions are very small, but there are also others in which they extend over the entire diameter of the conduits. It is clear that these latter conditions, which cause the appearance of multiphase structural scales intermingled with the scales of the device being considered, and thus of the flow occurring, exceed the simple situation of very small heterogeneities. Even though these conditions complicate modeling, we must, nevertheless, in our theoretical description, position ourselves so as to address them. The methodology that we have chosen gives us a good perspective from which to approach this problem. The first models resulting from it, which are relatively classical in form, are not immediately well adapted to these situations but we will see how to improve them in this regard.

We have not only talked about multiphase structures but also about a certain degree of randomness; these are not well-ordered structures. The medium permanently displays disordered fluctuations of phase positions and velocity fluctuations, which very often resemble those existing in a turbulent flow. This situation originally occurs with the fact that we cannot completely control the preparation of the medium at the initial instant, or at entry of the flow field, either in terms of phases or in terms of the field of velocities. It is only when a relatively viscous liquid phase exists that this random character of preparation is not amplified during passage through the device. In the vast majority of cases, the initial disorder increases greatly during the flowing of the medium, especially when velocity is high; the spatial and temporal scales of the fluctuations grow to a certain level of saturation that ceases to be controlled solely by the initial and inlet conditions. Phenomena of amplification and saturation of fluctuations are in part the same as those that occur in single-phase turbulent flows. The multiphase character causes additional instabilities in the presence of trajectories of drops or particles (even in a Stokes flow, if more than two particles interact), and instabilities linked to the production or coalescence of drops or to collisions between solid particles enable the different inertias of the different phases to be fully at play. The resulting random and fluctuating medium can be referred to as an extended turbulent medium. It is similar to the classic single-phase turbulent medium, but shows fluctuations of new variables associated with additional phenomena and requires similar methods of modeling.

In addition to the averaged variables we have already mentioned, we will need to define and study the variances of fluctuations of these variables, and even their probability density functions, and find the balance equations they satisfy. In principle, these equations do not directly involve the physical nature of the phenomena that regulate these fluctuations; therefore, they can be used much more widely than for turbulent flows. The closure hypotheses of these equations will necessarily take into account the nature of the phenomena involved, and it is here that classic hypotheses for turbulence will have to be adapted. Moreover, it will also be necessary to take into account the fact that the small scales of fluctuations, which still retain the particularities of each type of medium being considered, may in this case retain an influence more often than in classic turbulent flows, where the so-called high Reynolds number situations are very common. Obviously, it will be necessary to complicate the representations of these fluctuating flows and the equations to which they lead, that is to build more complex models, but it will be interesting to be able to do this with the same spirit, and perhaps the same tools, as for classical turbulent flows.

We have also stated that various physical or chemical phenomena may occur, particularly, in the internal structure, giving it its specific utility. Momentum exchanges occur between various constituents via the intermediary of forces; exchanges of heat and mass occur as well (phase changes), most often coupled together. Chemical phenomena can also be used beneficially inside a given phase or at the interface between two phases. The approach we are seeking must be general enough to adapt to these diverse physical phenomena.

Part 1 of this book deals with the definition and description of the basic approach, which can be presented simply as the generalization to several phases concerning the usual approach used in the mechanics of continuous media to the thermomechanics of fluids in turbulent flow. Its objective is to provide a context of theoretical representation, its hypotheses, its possibilities and its limits, and to identify the different types of equations necessary to make this approach useful, i.e. to draw from it predictions concerning specific flows. The general description includes four aspects: the theoretical description of a piecewise continuous flow; the definition of an averaged or filtered description, usable in practice, and it is statistical averages that will first be used here; the writing of balance equations for averaged quantities that can be chosen to represent the medium in its evolution; and the definition of the necessary constitutive laws, which include equations of state and laws of irreversible processes, and the problem of choosing these. These basic developments are the subject of Chapters 1–4.

Part 2 of the book begins by discussing the principles of modeling the effect of irreversible processes that are active within the flows. There are, in Chapter 5, the irreversible processes linked to exchanges between phases, masses, momentum or energy. We will show the significance of defining and studying exchanges in the form of exchange fluxes per unit of interface area. It is not possible to calculate these exchange fluxes without understanding the detail of the phenomena occurring on a small scale in the immediate vicinity of the interfaces, and it is necessary to suggest approximate models. The specificities of the different multiphase flows must explicitly be involved in this; we will not attempt to examine each interesting case in detail, or even a single interesting case, but will study a certain number of specific local exchange configurations and discuss how they can serve to construct the desired models.

Equally important, there are also additional dispersion fluxes inside a phase, for momentum or energy, that are due to fluctuations, often turbulent in nature, the modeling of which is discussed in Chapter 6. We will see that they may be modeled by generalizing approaches that are usual in single-phase turbulent flows, but there remains at least one major problem to be studied: the influence of the presence of phases on the spectrum of the characteristic length scales. We will also discuss simplification in which the mass of one phase is considered simply as diffusing in the overall multiphase mixture itself, in a manner analogous to a mixture of truly miscible gaseous or liquid components, without examining its own convection velocity in detail.

In Chapter 7, we then focus on the mean interface area per unit of volume, which plays a critical role in all of these questions related to exchanges between phases. The generalization of the approach for filtered averages, such as the large eddy simulation, is discussed in Chapter 8. This approach is, particularly, indicated for more precise study of the presence of large scales of multiphase or dynamic structures, which are avoided in statistical approaches. There is still a lot of work to be done in these areas. To date, models and their approximations have been presented and discussed in relation to the available experimental data. Chapter 9 reconsiders most of these questions with the general viewpoint of thermodynamics of irreversible processes, which will require a generalization of the classic thermodynamic approach. Classically used approximations will be reinforced for the most part, but there will also be interesting openings. Chapters 10 and 11 focus on the experimental methods and results available to date, and on the new aspects to be studied further, after a brief description of the current and new experimental methods.

Part 3 analyzes in more detail the granular flows and media, a type of multiphase media that has generated recent interest. In Chapter 12, fluidized beds, which are widely used in industry, are considered as a typical example, and we give a summary of the various models proposed and show how the developments in the preceding chapters are useful to explain them, as well as to emphasize the improvements that are necessary. In Chapters 13–15, we discuss an improvement for dense granular media in which lasting contacts between grains must be taken into account. There is, of course, a regime of flow in such a media, where only short collisions are occurring, but this is a limited case, when the flows are relatively rapid. At the other limit of small displacements, the media can be studied like elastic solid media, and between these two limits one or more other regimes of motion may exist due to the fact that the contacts between two or more grains may last for more or less time, small deformations and displacements of grains take place, and sudden sliding is possible locally. The multiphase approach is still possible, but it is necessary to consider that each grain is a different solid phase for which we know the constitutive laws, and the search for constitutive laws for a solid phase that would be the ensemble of all of the grains is the center of the modeling issue. These situations are at the limit of the domain of soil mechanics, and we will not go deep into this area. We will confine ourselves to demonstrate how the approach can lead to a form of model in which the stress tensor of the medium is given by a constitutive law in the form of a differential balance equation, like a kind of viscoelastic medium with a threshold.

Part 4 examines the improvement of the representation of all these media and adds the possibility of better understanding the fluctuations around mean values. The fact that we have defined mean variables from the beginning does not actually mean that it is impossible to understand the fluctuations around them. This is possible, first, by simply studying the variances of the fluctuations of variables, which has already been discussed in the preceding chapters for velocity fluctuations. This will now be extended to enable knowledge of the probability density functions, even joint probability density functions, of the various variables in certain phases, which is an interesting improvement in several situations. When relatively rapid chemical reactions occur in a gas in the vicinity of liquid droplets that supply one of the reactants, it is absolutely necessary to take into account the strong and fluctuating heterogeneities of composition and temperature that are found in the gas, precisely due to the exchange of chemical species between phases and the chemical reactions locally occurring. In problems where the vaporization of an ensemble of drops takes place, the distribution of temperatures of different drops, if wide enough, can have a significant effect on the mean rate of vaporization. Finally, for the dispersion of particles or droplets, a better understanding of the distribution of fluctuations of velocities and diameters of droplets is very useful.

In Chapter 16, we examine the modeling of the probability density of compositions for the gas phase of a two-phase medium in combustion. Chapter 17 discusses the possibility of better understanding the temperature fluctuations of the liquid phase or of drops or solid particles, which can have significant consequences for phase changes; finally, in Chapter 18, we study the possibility of better understanding the distribution of velocity fluctuations and parcel sizes of a phase. To obtain a better understanding of the distribution of sizes of a phase, we must define different categories of parcels in this phase, with each one behaving like a phase itself, and the treatment of all these virtual phases can thus become quite unwieldy. For situations in which the phases are always highly dispersed, the classically used stochastic Lagrangian–Eulerian simulation method is more appropriate. We will show how this approach connects to the Eulerian method discussed in this book.

Since the field contains problems that are still confined largely to the research level (not only in the last part), we have been careful throughout to emphasize the main subjects in which modeling still has critical flaws and that must be the subject of the most promising effort.

Literature on the domain, less general and more focused on one of the applicative aspects, provides additional information and interesting developments. In particular, we will refer more than once to the following works: [SOO 89, DEL 81, NIG 91, OES 06, CRO 01, JAK 08, PRA 06, AND 11].

Finally, the basics of the thermo mechanical description of continuous media used here were taken from [BOR 08a] and [BOR 08b].

PART 1

Approach and General Equations

Part 1 of this book deals with the definition and description of the basic approach, which can be simply presented as the generalization to several phases of the usual approach of continuous media in the thermomechanics of fluids in turbulent flows. We thus often speak of “piecewise continuous media”. Its objective is to give a framework for the theoretical representation, its hypotheses, possibilities and also its limits, to identify the different types of equations that are necessary to make this approach useful, i.e. by taking the “predictions” concerning specific flows. It should be noted that this approach is the only one that allows us to provide exact, instantaneous equations of the complete medium, from which we can then develop equations (which require closure assumptions, such as the more classical equations for turbulent flows with a single phase) of variances and probability density functions (PDF). The general description consists of four aspects, which are each the object of a chapter: the theoretical description of a “piecewise continuous flow or medium”; the definition of an averaged or filtered description, usable in practice, and in particular the statistical averages that will be mainly used here; the writing of balance equations for averaged quantities that can be chosen to represent the medium in its evolution; and finally, the definition of the necessary constitutive laws, which include “equations of state” and laws of irreversible processes, and the problem of choosing them.

Chapter 1

Towards a Unified Description of Multiphase Flows

1.1. Continuous approach and kinetic approach

In classic fluid mechanics theories, fluid is usually considered as a “continuous medium”, described locally and at each instant using a certain number of characteristic variables, and its evolution is represented by “balance equations”. These balance equations are partial differential equations in three-dimensional space and time, whose original writing uses the Eulerian point of view: a geometric point in space is designated, and at this point, the characteristics of the fluid at each instant are observed, in particular its velocity. This leads to the so-called continuity equation and then to Euler and Navier–Stokes equations and necessitates the definition of the Cauchy stress tensor, containing the pressure and tensor of viscous stresses. The description is, therefore, not complete until these new variables can be given by specific “constitutive laws”, which represent the nature and small-scale properties of the fluid in question. These constitutive laws are not necessarily based solely on a theory; they can, with more generality, be of empirical origin, but in any case they must follow the principles of thermodynamics.

There is another way to find these same equations and laws by considering the fluid medium from the beginning as a set of atoms or molecules in steady motion and colliding frequently in the void space, under the action of the laws of classic mechanics for material points. This approach, often called the kinetic approach, was introduced by Boltzmann and Maxwell and has been used to recover classical Eulerian continuity equations and Navier–Stokes equations since the work of Chapman and Enskog in the early 20th Century. In this context, the characteristic variables that we might call “macroscopic” variables are defined in a small volume around each point in space, and in a small interval of time around a given instant. When the volume and the interval of time considered are very small in relation to the spatial and temporal scales of variation expected for macroscopic variables, in the macroscopic experiments and situations with which we are concerned, these macroscopic variables no longer depend on the real size of the volume nor the time interval, and have continuity properties with respect to space and time. It is in these conditions that we can consider the medium as continuous, on the macroscopic scale and with our limited view.

The base equation of the kinetic approach is the Boltzmann equation, which concerns the distribution function of the velocities of molecules at a given point M(x) and a given instant t. To be exact, and for a gas medium containing only one kind of molecule, this function, f(u, x, t), is defined as the number of molecules that can be found, in the medium, in a small volume around point x approximately dx, at an instant t approximately dt, and which have velocity u approximately du (here, of course, the variables x and u are three-component vectors, as are dx and du). The number of molecules per unit of volume of the medium is, thus, defined as the integral in space of the values of velocities (from minus infinity to plus infinity) of this distribution function, and what we might call the macroscopic velocity of the medium is the integral, in this same space of velocities, of the product uf, divided by the integral of f itself. All the molecules of the gas being identical and of known mass, the volumetric mass of the medium is the product of the number of molecules per unit of volume and the mass of a molecule, and the velocity of the medium thus appears as the ratio between the momentum of the medium and its mass, both per unit of volume. The historical development of this approach is discussed in [CHA 60] and a very complete recitation of the developments in [HIR 54]. In the framework of the approximation of continuous media, this approach allows us to find the same forms of the balance equations of mass and momentum by using the first continuous approach discussed above. However, in addition, the fact that we can describe in more detail the microscopic structure of the material enables us to obtain the constitutive laws in a theoretical manner, that is expressions of all the variables that appear in the balance equations, namely the Cauchy stress tensor, pressure, viscous stress tensor, etc. These expressions are calculable only because of certain hypotheses on the microscopic characteristics of the gas being considered (for example the approximation of molecules as hard spheres of a certain mass). The energy balance equation of the continuous medium can also be found by defining the heat conduction flow in particular. When the approximation of the continuous medium is no longer valid, as is the case for rarefied gases, the Boltzmann equation remains valid and the balance equations of mass, momentum and energy still exist with the same overall form, but they contain additional terms, and the usual terms can no longer be calculated using the same approximations. In certain conditions, it is no longer useful to define a macroscopic velocity for the medium, and the calculation of the velocity distribution function itself, by using adequate approximations to represent collisions between molecules in the Boltzmann equation, is recommended.

When undertaking a theoretical study of multiphase fluid media, we might ask ourselves which of the two approaches is of more interest, especially when we wish to focus on a case in which one of the phases is dispersed into numerous parcels of different sizes.

In fact, it seems that the first studies involving clouds of drops in a gaseous medium used the kinetic approach to study the liquid phase composed of the set of drops by examining a distribution function where the variables were both the velocities and sizes of the drops, or f(u, v, x, t), where v is the volume of the drops. At a given point and at a given instant, the number per unit of volume (of the medium) of drops of volume v, approximately dv, is still the integral of f on the space of velocities, and the total number of drops of all sizes is the integral of f both on the space of velocities and that of volumes. The function f follows a type of Boltzmann equation here again, in which the most difficult term to approximate is the term due to the collisions of particles. We may refer, for example to, [WIL 58]. However, it has never been suggested to represent the gaseous phase in which these drops are dispersed using a Boltzmann equation that, when the quantity of drops is relatively large, might include the influence of these drops on the motion of the fluid. The description of this gaseous phase has been kept in the form of a continuous medium, where the influence of the drops is represented by additional interaction terms.

For less well-defined multiphase media such as those containing bubbles and gas pockets that can be of various shapes, a “macroscopic continuous medium” approach has been applied from the beginning, by focusing on the characteristic variables for a small volume around each point in space. The basic hypothesis has been made that this volume is very large in relation to the size of the inclusions of the dispersed phases, and these variables therefore represent an “equivalent continuous medium” corresponding to our multiphase medium. The fictive volume within which the macroscopic variables are defined is often called the “representative elementary volume”. We may refer, for example to, [SOO 89] and [DEL 81]. In these conditions, the balance equations for these variables have the classic form but the necessary constitutive laws are more complicated; they cannot be calculated and must be obtained through experimentation. This approach is valid, but the definition of the characteristic variables of the medium, which must have a certain continuity, as well as those of the various constitutive laws, must be precisely made. Obtaining these laws is (too) highly dependent on experimentation, and the domain of validity of these laws is difficult to know with precision. Moreover, the condition that each inclusion must be much smaller than the representative elementary volume has very often been invalidated, or noted visually as soon as modern viewing methods were able to be used. Does this flaw seriously affect the approach, or is it responsible for only “small” errors? This remains an ongoing debate and, most likely, must depend on each of the various cases considered.

More recently, in the 1980s, it appeared possible to develop an approach that might be called intermediary, which defined the equivalent continuous medium more accurately and thus made it possible to obtain more information about the form of the constitutive laws, which in turn made it possible, importantly, to better examine situations with large inclusions, and could therefore be adapted to widely diverse practical situations. In fact, if the size of each inclusion in a phase is large enough to include a large number of molecules that interact on the interface with extremely numerous molecules of the other phase in contact, then it is possible to treat all of the phases, fluid or solid, like classic continuous media. The multiphase medium is, thus, a “piecewise continuous medium” in which each piece of continuous medium is separated from the other by an interface where the properties can show a discontinuity. Using this perspective, we enable the shape and size of the inclusions in each phase to vary in the medium in a totally arbitrary manner. However, the balance equations for phases in continuous media are unusable as such for representing the full multiphase medium, for the same reason that equations of material points mechanics cannot be used to represent a gas, which is a group of molecules. We do not precisely know the initial conditions of each molecule in the gas, nor the initial position or shape of each inclusion in each phase, even if these inclusions are not too small. To establish some macroscopic characteristics at the scale of a multiphase medium, and to deduce the macroscopic equations that they must satisfy, it is necessary to define, through an averaging operation that will be specified later, a “mean continuous” medium that will be the equivalent continuous medium. This approach has grown slowly since its introduction. It was presented for the first time in [DRE 83]. However, it is very similar to the approach used for flows in porous media by Marle in the beginning of 1967 [MAR 67]. At roughly the same time, Nigmatulin began using the same approach in the foundation chapters of his book [NIG 91], which also presents highly useful and detailed developments for several different applications. General equations for piecewise continuous media have been studied in detail by Kataoka [KAT 86], taking into account mass exchanges as well as exchanges of momentum or energy between these phases. This generality is one of the strong points of the approach.

This averaged piecewise continuous medium approach is the approach that we will follow in this book. In this way, we will avoid defining a distribution function for the sizes and velocities of parcels, but it will be important to know other macroscopic variables of the medium besides its volumetric mass and velocity. We will be interested at least by the “volume fraction” occupied in the medium by each of the phases, and also by the mean temperature or internal energy of each phase, which are to be defined precisely. It will also be necessary to have a thorough understanding of at least one macroscopic variable related to the size of the inclusions of a phase that is dispersed in another carrier phase. First, and in the most general of situations, it is through the “mean area of interface per unit of volume” of the medium that this knowledge will be introduced and studied. For the case of dispersed phases with “particles” of various sizes, if necessary, we can also understand the distribution of sizes and velocities by defining and studying the number of particles of volume v (approximately dv). This will come down to considering a multiphase medium with an infinite number of phases, with each category of particle in the given volume able to be considered as a different phase.

In Chapter 2, we will describe the basic equations that represent a piecewise continuous medium instantaneously and locally. The definition of averages and the obtaining of equations related to equivalent continuous media will be discussed in the following chapters.

1.2. Eulerian–Lagrangian and Eulerian formulations

If we wish to study only situations in which, in a fluid, small quasi-spherical particles of another fluid or a solid are dispersed, one practical method that comes immediately to mind is simply to study the motion of the center of gravity of these particles, taking into account the friction between them and the fluid that surrounds them, which depends on their volume and shape, using specific formulas that experiments can validate. If it is not only their motion that interests us, but also their temperature, for example, we can also study this quantity by supposing that it is homogeneous in the particle in question. This brings us back to consider the total internal energy of the particle, and its dependence on exchanges of heat and possibly of mass with the outside environment, without necessarily having to specify whether these changes occur more on one side or the other side of the particle, and using more or less empirical formulas.

This point of view, which separates the carrier fluid, still considered as a continuous medium, from a certain number of isolated objects, has been highly developed. It is not too complicated to implement when the particles have a well-defined geometric shape and retain this shape permanently. It has been used to construct an Eulerian representation of a mean particulate medium (for example [JAC 00]) and also, especially, for the so-called Eulerian-Lagrangian approach, in which the continuous fluid medium is still described by Eulerian equations but the particles of the other phase are observed in a Lagrangian manner (see [OES 06]). Suspensions containing liquid drops or solid particles can be studied using this perspective, as long as the droplets or particles do not break apart or cluster together too much. Exchanges between particles and the continuous medium have been studied in great detail, and these studies constitute a body of knowledge that is quite generally useful; we recommend, for example, [SIR 92] and [SIR 99]. The book by Crowe also provides a broad recapitulation of the knowledge and practices using this approach [CRO 01].

This particular point of view is not easy to generalize for types of multiphase media other than highly dispersed environments, and it is not inclined to allow a general theoretical approach, not least because it immediately introduces asymmetry into the description. Moreover, Lagrangian–Eulerian representation introduces significantly greater difficulties of application in situations where, as the number of particles is higher, these particles interact more with each other than with the fluid, and in situations where these particles do not have a constant and simple shape. There are, of course, methods for calculating the Lagrangian motion of solid parcels for granular media, taking into account the multiple contacts between these “grains”, but these contacts are firmly localized; the fluid between the grains is not taken into account, and increasing the number and complicating the shape of the grains greatly increases the calculation time. The Lagrangian point of view has certain advantages, not the same as the Eulerian point of view does; it is beneficial to make use of these advantages, but the theoretical separation of a Lagrangian part and an Eulerian part has no general justification.

The unified approach we are looking for will be completely Eulerian: the macroscopic continuous medium equivalent to the multiphase medium, which is non-continuous by nature, will be described by Eulerian balance equations. The necessary models will, therefore, be studied first of all in a Eulerian context. Nevertheless, it is of interest, as we will see in Chapter 18, to use a Lagrangian translation of certain equations and models, which is closer to the Lagrangian–Eulerian representation, often used in practice. We will then show how to link the balance equations of the two approaches. In addition, the possibility of coupled calculations, in the various geometric zones of the space, between a model expressed in Eulerian form and a model expressed in Lagrangian–Eulerian form is also a promising practical solution.

Chapter 2

Instant Equations for a Piecewise Continuous Medium

If we consider any multiphase medium at a given instant in any type of experimental situation, this medium is a “piecewise continuous” medium with different non-miscible phases because the local density of the medium undergoes sudden jumps on certain surfaces in the medium. Other local characteristics per unit of mass can also display these kinds of discontinuities, but it is not always the case. We will discuss this further later on. Here, we are considering a medium with two or more phases, at least one of which is fluid (gas or liquid); the others can be solid or fluid. A phase is usually defined by its physical nature, but it is also possible to define phases using other criteria, provided that several phases cannot be in the same place at the same instant; we will return to this point later as well.

2.1. Integral and differential forms of balance equations

The simplest form of a balance equation concerns a given control volume of space, defined by the experimentalist and set in relation to the chosen reference coordinates. For each extensive variable of the medium (total mass, mass of a certain type, total momentum, momentum of mass of a certain type, total energy, etc.), we simply write that the temporal variation rate is either due to processes of exchange with the exterior environment via the surface of the volume, or due to phenomena of production (positive or negative) internal to the control volume. If we write the extensive variables per unit of mass that can be defined locally as g, we have:

The second term to the left of the equals sign is on the left because the normal to the surface of the volume is defined, as is usual, as the outgoing normal. This is the flux of quantity g exiting from the volume (and thus exchanged with the exterior), and we can write it formally by defining a velocity associated with this exchange flux per unit of surface in direction α. The quantity is the rate of production (or disappearance) of the quantity in question, per unit of volume, due to any physical or chemical process possible (which must then be identified). The dot above is simply a notation and does not indicate a temporal derivative; however, this notation refers to the fact that, without an exchange with the exterior, the previous balance equation shows that the rate is equal to a temporal derivative. The presence of the volume mass factor is simply a convention that is widely but not always used.

For a continuous medium, using the so-called divergence theorem, which transforms the surface integral into a volume integral, we deduce from an integral balance, for a fixed control volume of any shape and position, a balance partial differential equation, as follows:

We are mainly interested here in the classic balances of total mass, momentum and total energy, which we rewrite showing the fluxes and rates of production:

The mass balance of a substance j in the fluid medium is written, with the mass fraction of this substance, as:

The diffusion flux of this substance in the medium is related to the diffusion velocity previously defined: , and the term of production, or disappearance, of the substance in question is zero unless it is caused by chemical reactions.

These differential balance equations are valid in each phase of a multiphase medium, whether these phases are gaseous, liquid or solid, with differences in the physical meaning and expression of the values defined in each of these phases. Here, we have applied classic principles, stipulating that the production rate of the total mass is always zero, that the production rate of momentum is the sum of the remote forces and that the total energy production rate is equal to the power of the remote forces. It is supposed here that the remote forces are composed solely of gravity due to Earth, and thus are linked to gβ, but this is transposable to other types of forces.

Integral balances can also be written for an environment in which two or more phases are present in the volume, Vol. By using a phase index as stated above (1 in phase i and 0 elsewhere) Φi, we have:

[2.1a]

The production rate can include volumetric phenomena, as well as phenomena that occur only at the interface, and in this case, the part of the rate that represents this surface phenomenon has a Dirac peak structure placed at this interface (and its volumetric integral is therefore well defined).

A particularly interesting application of this balance concerns a very small volume that only contains a small portion of the interface between two phases i and i′ and then moves along with this interface. We can also write an integral balance equation for a mobile volume; and in this case, the second term has to use the velocity relative to the surface of the volume in motion. Here, the volume is moving with the interface, so we will have . When a very small volume is considered, the first term of the above equation is zero, the second term is simply the sum of the fluxes of g exiting from the side of phase i and the side of phase i′ and the third term is reduced to the integral of the rate of production at the interface. In this case, by defining as the normal at the interface exiting from as the normal exiting from ), we get the general algebraic equation valid at all points of the interface:

[2.1b]

The index s associated to i or i′ indicates that the quantities within the phase are taken immediately at the interface. The term on the right-hand side represents the overall production rate per unit of surface of the quantity considered by surface processes at the interface between phases. It is zero unless such processes can exist and must be provided by studying the involved physics.

For masses, momentum and total energy, in the classic case that does not consider surface tension, these surface terms are zero and this interface balance relationship is written simply as:

These relationships are used directly, leaving out the irreversible effects of viscosity and heat transfer, when we study shock waves or detonation or deflagration waves. We will see their general usefulness later on, as well as how to include the effects of surface tension in them.