Thermo-Fluid Dynamics of Two-Phase Flow

This book is intended to be a basic reference on the thermo-fluid dynamic theory of two-phase flow. The subject of two or multiphase flow has become increasingly important in a wide variety of engineering systems for their optimum design and safe operations. It is, however, by no means limited to today’s modern industrial technology, and multiphase flow phenomena can be observed in a number of biological systems and natural phenomena which require better understandings. Some of the important applications are listed below.

The singular characteristic of two-phase or of two immiscible mixtures is the presence of one or several interfaces separating the phases or components. Examples of such flow systems can be found in a large number of engineering systems as well as in a wide variety of natural phenomena. The understanding of the flow and heat transfer processes of two-phase systems has become increasingly important in nuclear, mechanical and chemical engineering, as well as in environmental and medical science.

The design of engineering systems and the ability to predict their performance depend on the availability of experimental data and conceptual models that can be used to describe a physical process with a required degree of accuracy. From both a scientific and a practical point of view, it is essential that the various characteristics and properties of such conceptual models and processes are clearly formulated on rational bases and supported by experimental data. For this purpose, specially designed experiments are required which must be conducted in conjunction with and in support of analytical investigations. It is well established in continuum mechanics that the conceptual models for single-phase flow of a gas or of a liquid are formulated in terms of field equations describing the conservation laws of mass, momentum, energy, charge, etc. These field equations are then complemented by appropriate constitutive equations such as the constitutive equations of state, stress, chemical reactions, etc., which specify the thermodynamic, transport and chemical properties of a given constituent material, namely, of a specified solid, liquid or gas.

The importance of the Eulerian time averaging in studying a single-phase turbulent flow is well known. Since the most useful information in analyzing standard fluid flow systems is the time mean values rather than the local instant responses of the fluid, its use both in experimental and analytical purposes is indispensable in turbulent flow studies. For example mean velocity, temperature and pressure or the heat transfer coefficient and the friction factor are the important mean values routinely required in standard problems. Furthermore, commonly used experimental methods and measurements are well suited for the application of the time average. Thus, a single-phase turbulent flow has been studied in great depth by using the time averaged field equations with the constitutive laws expressed by mean values. Although these models, which are based on time averaging, do not give answers to the fundamental origin, structure and transport mechanisms of turbulent flow, their applications to engineering systems are widely accepted as efficient means of solving problems.

In the preceding chapter, the important definitions and basic relations between them have been given. We now apply them to the time averaging of the balance laws in the two-phase flow media. As it has been explained in the Section 1.1 of Chapter 4, it was necessary to introduce several sets of time intervals because of the discontinuous changes in the nature of fluid surrounding the point of average. Thus the domain of averaging has been divided into [Δt]_S and [Δt]_T. During [Δt]_T, the standard balance equation (2–6) holds, since the fluid occupying the point x ₀ can be considered as a continuum. However, in [Δt]_S the interfacial balance equation, namely, the jump condition of the Section 1.2 of Chapter 2, is valid because the characteristics of the interface dominates in this time interval.

The time-mean values are consistently expressed by the spatial description as shown by the definitions (4–15) and (4–16), and the idea of the particle coordinates for the averaged two-phase flow fields is not clear nor trivial due to the phase changes and the diffusions. The phase change corresponds to the production or disappearance of fluid particles for each phase throughout the field. The difficulty arises because each phase itself does not apparently obey the corollary of the axiom of continuity, namely, the permanence of matter. However, the diffusion of each phase permits the penetration of mixture particles by other fluid particles. It is clear that the material coordinates, which is the base of the standard continuum mechanics, is not inherent to a general two-phase flow field obtained from the time averaging. However, it is possible to introduce mathematically special convective coordinates which are useful in studying the kinematics of each phase and of the mixture.

The exact forms of the interfacial transport terms I _k and I _m for mass, momentum and energy interchanges have been given in the Section 1.2 of Chapter 5. However, they are expressed by the local instant variables, thus it is not possible to use them as the constitutive laws in the averaged field equations. It is evident that we need to understand the physical meaning of these terms in detail before constructing any particular constitutive equations for two-phase flow systems. With this in mind we clarify different physical mechanisms controlling these terms as well as to identify important parameters on which they depend. Furthermore, it is important to accept that not all the characteristics inherent to the local instant two-phase flow can be brought into the time-averaged model. We consider that the averaged field equations express general physical principles governing the macroscopic two-phase flows while the constitutive equations approximate the material responses of a particular group of systems with simple mathematical models. In this connection, we make a number of assumptions in the interfacial transfer terms in order to both distinguish the dominant transfer mechanisms and also eliminate some of the complicated terms that have insignificant effects in the macroscopic field.

The two-fluid model (Ishii, 1975, Ishii and Mishima, 1984) is formulated by considering each phase separately. Thus, the model is expressed in terms of two sets of conservation equations governing the balance of mass, momentum and energy in each phase. However, since the averaged fields of one phase are not independent of the other phase, we have interaction terms appearing in these balance equations. The terms denoted by Г _k, M _k and E _k are the mass, momentum and energy transfers to the k ^th-phase from the interfaces. As these quantities also should obey the balance laws at the interfaces, we have derived the interfacial transfer conditions from the local jump conditions. Consequently six differential field equations with three interfacial transfer conditions govern the macroscopic two-phase flow systems.

The interfacial transfer terms are strongly related to the interfacial area and to the local transfer mechanisms, such as the degree of turbulence near the interfaces and the driving potential. Basically, the interfacial transport of mass, momentum and energy is proportional to the interfacial area concentration and to a driving force. This area concentration, defined as the interfacial area per unit volume of the mixture, characterizes the kinematic effects; therefore, it must be related to the structure of the two-phase flow. The driving forces for the interphase transport characterize the local transport mechanism and they must be modeled separately.

The two-fluid model is widely used in the current two-phase flow analysis codes, such as nuclear reactor safety analysis codes RELAP5, TRAC, and CATHARE. In the conventional model, the interfacial area concentration is given by empirical correlations. The correlations are based on two-phase flow regimes and regime-transition criteria that do not dynamically represent the changes in interfacial structure. There exist the following shortcomings caused by this static approach.

In analyzing the interfacial force and relative motion between phases, consider first, the momentum equation for each phase. Under the assumption that both the average pressure and stress in the bulk fluid and at the interface are approximately the same, the k-phase momentum equation is given by where , and M _ik are the average viscous stress tensor, the average turbulent stress tensor, the interfacial shear stress, and the generalized interfacial drag force. The conservation of the mixture momentum requires which is the modified form of the average momentum-jump condition. Constitutive equations of the average turbulent stress tensor and the generalized interfacial drag force are required to analyze two-phase flows using the two-fluid model.

The basic concept of the drift-flux model is to consider the mixture as a whole, rather than two phases separately. It is clear that the drift-flux model formulation will be simpler than the two-fluid model, however it requires some drastic constitutive assumptions causing some of the important characteristics of two-phase flow to be lost. However, it is exactly this simplicity of the drift-flux model that makes it very useful in many engineering applications. As it is the case with the analyses of two-phase flow system dynamics, information required in engineering problems is often the response of the total mixture and not of each constituent phase (Tong, 1965). Furthermore, detailed analyses on the local behavior of each phase can be carried out with less difficulty, if these mixtures responses are known.

Two-phase flow always involves some relative motion of one phase with respect to the other; therefore, a two-phase-flow problem should be formulated in terms of two velocity fields. A general transient two-phase-flow problem can be formulated by using a two-fluid model or a drift-flux model, depending on the degree of the dynamic coupling between the phases. In the two-fluid model, each phase is considered separately; hence the model is formulated in terms of two sets of conservation equations governing the balance of mass, momentum, and energy of each phase. However, an introduction of two momentum equations in a formulation, as in the case of the two-fluid model, presents considerable difficulties due to mathematical complications and uncertainties in specifying interfacial interaction terms between two phases (Delhaye, 1968; Vernier and Delhaye, 1968; Bouré and Réocruex, 1972; Ishii, 1975). Numerical instabilities caused by improper choice of interfacial-interaction terms in the phase-momentum equations are common. Therefore, careful studies on the interfacial constitutive equations are required in the formulation of the two-fluid model. For example, it has been suggested (Réocruex, 1974) that the interaction terms should include first-order time and spatial derivatives under certain conditions.

The two-fluid model is the most detailed and accurate macroscopic formulation of the thermo-fluid dynamics of two-phase systems. In the two-fluid model, the field equations are expressed by the six conservation equations consisting of mass, momentum and energy equations for each phase. Since these field equations are obtained from an appropriate averaging of local instantaneous balance equations, the phasic interaction term appears in each of the averaged balance equations. These terms represent the mass, momentum and energy transfers through the interface between the phases. The existence of the interfacial transfer terms is one of the most important characteristics of the two-fluid model formulation. These terms determine the rate of phase changes and the degree of mechanical and thermal non-equilibrium between phases, thus they are the essential closure relations that should be modeled accurately. However, because of considerable difficulties in terms of measurements and modeling, reliable and accurate closure relations for the interfacial transfer terms are not fully developed. In spite of these shortcomings of two-fluid models, there is, however, no substitute available for modeling accurately two-phase phenomena where two phases are weakly coupled. Examples of these are: