Introduction to Multiphase Flow

This chapter introduces the reader to multiphase flows and to phenomena that are unique to them. A special but most common case of multiphase flows are the two-phase, gas-liquid flows. Multiphase flows are present whenever there is heat transfer accompanied with phase change such as boiling and condensation. The unique nature of multiphase flows is made evident by a few examples. The definitions and the bases necessary for dealing with such flows are given: the various definitions of the void fraction (generically speaking, the fraction of space or time occupied by the gas phase), the phase flow rates, velocities, etc., and the flow quality (the ratio between the mass flow rate of the gas to the total mass flow rate). The void fraction has a major influence on the flow regimes that characterize the topological arrangement of the two phases in the flow channel. Averaging of the flow properties in time and/or space is usually needed to deal analytically with multiphase flows. The various types of averaging needed are reviewed. The cross-sectional-average velocities of the two (or more) phases are usually not equal. The cross-sectional-average void fraction, the ratio of the average phase velocities and the quality are linked by a so-called triangular relationship. In the case of homogeneous flow, the phase velocities are assumed to be equal. The difference between the actual flow quality and the thermal-equilibrium quality is clarified. A few non-dimensionless numbers or groups of variables that are very often used are introduced in anticipation of their actual use in the following chapters. The most often used sources of information on multiphase flows are listed at the end of the chapter. Appendices at the end of the volume provide a tutorial for the reader not familiar with some needed notions of fluid mechanics and heat transfer, define the usual nomenclature and provide conversion factors.

The general methods of solution of thermal-hydraulic problems are recalled first and then we show how these are complicated by the presence of multiphase flows, before entering into the descriptions of the various approaches commonly used. The special features of multiphase or two-phase flows are pointed out, in particular the existence of a number of flow regimes. The various two-phase flow and boiling heat transfer variables of interest, such as the pressure gradient, the void fraction, the heat transfer coefficient, etc., will depend on the particular flow regime. In principle, one should model each flow regime separately; when flow-regime-specific models are used, one can “mechanistically” take into consideration the particularities of each regime. The alternative approach often used is to largely ignore the flow regimes and derive methods (most often empirical correlations) covering all flow regimes continuously. The complete formulation of the two-phase-flow problem, which would have required the description of the evolution in time of the fields (pressure, velocity, temperature, etc.) for each phase, together with a prediction of the geometry of the interfaces, is generally impractical. The often chaotic flow fields must be treated in terms of statistical, average properties. There are two general approaches, the two-fluid, or more generally the multi-fluid approach and the mixture formulation. A simple presentation of the two-fluid approach is given. The basis of the method is to write conservation equations for each phase and to include in these equations terms which represent the interaction between the phases. The closure laws required to complete this formulation are listed and examples of implementation difficulties are given. The phase conservation equations may be summed up to yield mixture conservation equations, a particular case is the homogeneous flow model. The relatively new developments that rely on computational fluid mechanics methods to analyse and simulate multi- and two-phase flows are introduced to the reader.

Interfacial instabilities govern many phenomena in two-phase flows, including the stability of jets, bubbles, droplets, films, etc. Following a brief introduction of the basic interfacial stability mechanisms and types of instabilities, stability criteria for the two main types of interfacial instabilities, namely the Rayleigh-Taylor and the Kelvin-Helmholz instabilities are derived. Applications to various cases such as the stability of films in boiling, stability of the dispersed phase (droplets or bubbles) in two-phase flow, etc. follow. The chapter also mentions the various extensions of the basic theory presented to a multitude of more complex situations, governed, however, the same general principles.

The chapter begins by describing the various two-phase flow regimes observed in pipes. Flow regime or pattern maps are used to identify the flow regime in terms of flow variables. Selected empirical flow regime maps are presented and the ways in which these may be generalized is discussed. A discussion of the analytical description of some important individual flow-pattern transitions follows. Flow regime maps can be analytically put together by using a set of transition criteria between flow regimes. The generation of full sets of criteria that can be used to build flow-pattern maps is described. Most commonly used flow regime maps for horizontal, vertical and inclined pipes are discussed at some length. This chapter is very extensive and long as it goes well beyond the description of the flow regimes and introduces the mechanistic methods used to define the transitions between regimes and their assembly into analytical flow regime maps.

The chapter begins by describing the principles of the various methods available for measuring the void fraction; these include photographic techniques optical or electrical techniques that produce point measurements of the void fraction, etc. One can obtain point measurements, chordal averages, cross-sectional averages or volume averages in the two-phase flow duct by various means. The next sections move into prediction methods for void fraction. In the case of the simplest possible model of two-phase flow, the homogeneous model, the situation is very simple and analytical expressions are readily available. For separated flows where the two-phase flow at different average velocities, a large number of void fraction correlations have been proposed over the years; the most often used ones are presented. The Drift-Flux model is an excellent analytical framework for dealing in general with two-phase flows and for estimating in particular the void fraction. Following the basic derivation of the model, the physical meaning of its two constants is discussed and a number of expressions are presented that allow computation of the void fraction, the phase velocities and other variables of interest in terms of the drift-flux parameters. The extraction of the Drift-Flux model parameters from experimental data is then discussed. A section is then devoted to the discussion of various proposals made regarding the Drift-Flux model parameters. One of the latest Drift-Flux model correlations, by Chexall-Lellouche is presented, followed by a section on comparisons of the performance of the various correlations. Finally, the void fraction in horizontal or near-horizontal flows of particular interest to the petroleum industry is discussed.

Void fraction is a quantity of primary importance in the design and analysis of two-phase systems. The pressure gradient in two-phase flow is present in the momentum conservation equation; at steady state, it has three components, the gravity, acceleration and friction terms. The first two can be computed in a straightforward way if the void fraction distribution in the duct is known. The frictional pressure gradient requires further modelling and the largest part of the chapter is devoted to this. Traditionally, the frictional pressure gradient (or frictional pressure drop) in two-phase flows has been obtained using a reference pressure gradient (based on the flow in the pipe of one of the phases alone) corrected for the presence of two-phase flow by a multiplier. Martinelli introduced this method that is still widely used today long time ago.

The simplest way to compute the pressure drop in two-phase flow is when one uses the homogeneous model but this is generally not sufficiently accurate. A number of the numerous correlation methods that have been proposed over the years for the frictional multiplier are reviewed and the most often used ones discussed. Comparisons about the quality of the predictions by various correlations are presented. At the end of the chapter, there is a special section devoted to the methods used in the petroleum industry for large pipes. There is also a brief general discussion on the pressure drop in piping singularities.