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...