Thermo-Fluid Dynamics of Two-Phase Flow

Frontmatter

Chapter 1. Introduction

Abstract

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.

Mamoru Ishii, Takashi Hibiki

Chapter 2. Local Instant Formulation

Abstract

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.

Mamoru Ishii, Takashi Hibiki

Chapter 3. Various Methods of Averaging

Abstract

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.

Mamoru Ishii, Takashi Hibiki

Chapter 4. Basic Relations in Time Average

Abstract

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.

Mamoru Ishii, Takashi Hibiki

Chapter 5. Time Averaged Balance Equation

Abstract

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.

Mamoru Ishii, Takashi Hibiki

Chapter 6. Connection to Other Statistical Averages

Abstract

The basic concept of the Eulerian statistical averaging has been explained in Chapter 3. By considering a set of N similar samples or systems, a statistical mean value is defined by a simple arithmetic mean among them, Eq.(3-7). Thus, the mathematical operation of integration for the time averaging should be replaced by that of summation in the statistical averaging.

Mamoru Ishii, Takashi Hibiki

Chapter 7. Kinematics of Averaged Fields

Abstract

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.

Mamoru Ishii, Takashi Hibiki

Chapter 8. Interfacial Transport

Abstract

The exact forms of the interfacial transport terms k I and m I 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.

Mamoru Ishii, Takashi Hibiki

Chapter 9. Two-fluid Model

Abstract

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.

Mamoru Ishii, Takashi Hibiki

Chapter 10. Interfacial Area Transport

Abstract

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.

Mamoru Ishii, Takashi Hibiki

Chapter 11. Constitutive Modeling of Interfacial Area Transport

Abstract

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.

Mamoru Ishii, Takashi Hibiki

Chapter 12. Hydrodynamic Constitutive Relations for Interfacial Transfer

Abstract

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.

Mamoru Ishii, Takashi Hibiki

Chapter 13. Drift-Flux Model

Abstract

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.

Mamoru Ishii, Takashi Hibiki

Chapter 14. One-Dimensional Drift-Flux Model

Abstract

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-phaseflow 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.

Mamoru Ishii, Takashi Hibiki

Chapter 15. One-Dimensional Two-Fluid Model

Abstract

The two-fluid model is the most detailed and accurate macroscopic formulation of the thermo-fluid dynamics of two-phase systems. In the twofluid 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.

Mamoru Ishii, Takashi Hibiki

Chapter 16. Two-Fluid Model Considering Structural Materials in a Control Volume

Abstract

The local instantaneous formulation of the differential balance equation has been well described by several authors (Ishii, 1975; Ishii and Hibiki, 2006; Drew and Passman, 1998). These equations suffer from various difficulties mainly due to the existence of the interface.

Mamoru Ishii, Takashi Hibiki

Chapter 17. One-Dimensional Interfacial Area Transport Equation in Subcooled Boiling Flow

Abstract

The one-dimensional form of the interfacial area transport equation can be obtained by applying cross-sectional area averaging over the threedimensional form of the interfacial area transport equation. However, the exact mathematical expressions for the area-averaged sink and source terms would involve many covariances that might further complicate the onedimensional problem.

Mamoru Ishii, Takashi Hibiki

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