One-dimensional drift-flux model and constitutive equations for relative motion between phases in various two-phase flow regimes

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Abstract

In view of the practical importance of the drift-flux model for two-phase flow analysis in general and in the analysis of nuclear-reactor transients and accidents in particular, the kinematic constitutive equation for the drift velocity has been studied for various two-phase flow regimes. The constitutive equations that specify the relative motion between phases in the drift-flux model has been derived by taking into account the interfacial geometry, the body-force field, the shear stresses, the interfacial momentum transfer and the wall friction, since these macroscopic effects govern the relative velocity between phases. A comparison of the models with existing experimental data shows a satisfactory agreement.

Introduction

Two-phase flows always involve 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 [1], [2] or a drift-flux model [3], [4], 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 because of mathematical complications and of uncertainties in specifying interfacial interaction terms between two phases [1], [2]. 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.

These difficulties associated with a two-fluid model can be significantly reduced by formulating two-phase problems in terms of the drift-flux model, in which the motion of the whole mixture is expressed by the mixture momentum equation and the relative motion between phases is taken into account by a kinematic constitutive equation. Therefore, the basic concept of the drift-flux model is to consider the mixture as a whole, rather than as two separated phases. The formulation of the drift-flux model based on the mixture balance equations is simpler than the two-fluid model based on the separate balance equations for each phase. The most important assumption associated with the drift-flux model is that the dynamics of two phases can be expressed by the mixture-momentum equation with the kinematic constitutive equation specifying the relative motion between phases. The use of the drift-flux model is appropriate when the motions of two phases are strongly coupled.

In the drift-flux model, the velocity fields are expressed in terms of the mixture center-of-mass velocity and the drift velocity of the vapor phase, which is the vapor velocity with respect to the volume center of the mixture. The effects of thermal non-equilibrium are accommodated in the drift-flux model by a constitutive equation for phase change that specifies the rate of mass transfer per unit volume. Since the rates of mass and momentum transfer at the interfaces depend on the structure of two-phase flows, these constitutive equations for the drift velocity and the vapor generation are functions of flow regimes.

The drift-flux model is an approximate formulation in comparison with the more rigorous two-fluid formulation. However, because of its simplicity and applicability to a wide range of two-phase-flow problems of practical interest, the drift-flux model is of considerable importance. In particular, the model is useful for transient thermohydraulic and accident analyses of both LWR’s and LMFBR’s [4]. In view of the practical importance of the drift-flux model for two-phase-flow analyses, the constitutive equations that specify the relative motion between phases in the drift-flux model have been derived by Ishii [4] by taking into account the interfacial geometry, the body-force field, shear stresses, and the interfacial momentum transfer, since these macroscopic effects govern the relative velocity between phases. To derive the simplified one-dimensional constitutive equations specifying the one-dimensional drift velocity, Ishii [4] assumed that the average drift velocity due to the local slip can be predicted by the same expression as the local constitutive relations, provided the local void fraction and the difference of the stress gradient are replaced by average values. The validity of the constitutive equations developed by Ishii [4] has been supported by various experimental data over various flow regimes and a wide range of flow parameters. However, it is anticipated that such simplified one-dimensional constitutive equations may not give a good prediction in some extreme flow conditions such as very high liquid flow and microgravity conditions where the effect of the wall shear stress on the relative velocity between phases may become significant. From this point of view, the purpose of the present study is to derive more rigorous constitutive equations specifying the one-dimensional drift velocity by taking into account the wall shear stress. The derived constitutive equations are validated by existing experimental data.

Section snippets

One-dimensional drift-flux model

Averaging over the cross-sectional area is useful for complicated engineering problems involving fluid flow and heat transfer, since field equations can be reduced to quasi-one-dimensional forms. By area averaging, the information on changes of variables in the direction normal to the mean flow within a channel is basically lost; therefore, the transfer of momentum and energy between the wall and the fluid should be expressed by empirical correlations or by simplified models. The rational

Relative motion in single-bubble system in confined channel

Recently, Tomiyama et al. [6] derived the relative velocity in a confined channel by taking into account the effect of a frictional pressure gradient due to a liquid flow. In what follows, we shall summarize the result in a simple form useful for the development of the drift constitutive equation in multi-bubble systems.

By denoting the relative velocity of a single bubble in an infinite medium by vr∞=vgvf∞, we define the drag coefficient byCD≡−2FDρfvr|vr|πrb2,where FD and rb are the drag

Dispersed two-phase flow

Ishii [4] developed a simple correlation for the distribution parameter in bubbly-flow regime. Ishii first considered a fully developed bubbly flow and assumed that the distribution parameter would depend on the density ratio, ρg/ρf and on the Reynolds number, Re defined by 〈jfD/νf. As the density ratio approaches unity, the distribution parameter should become unity. Based on the limit and various experimental data in fully developed flows, the distribution parameter was given approximately by

Example computation of newly developed one-dimensional drift velocity

In this section, an example computation of the newly developed constitutive equations for one-dimensional drift velocity will be performed for dispersed two-phase flows. The figure at the upper left of Fig. 1 shows an example computation of the newly developed constitutive equations for one-dimensional drift velocity in the 〈jf〉 vs. 〈〈Vgj〉〉 plane. The computational conditions are D=10.0 mm and 〈αg〉=0.10. The value of D=10.0 mm is about the hydraulic equivalent diameter of the flow channel in 17 ×

Conclusions

In view of the practical importance of the drift-flux model for two-phase flow analysis in general and in the analysis of nuclear-reactor transients and accidents in particular, the distribution parameter and the drift velocity have been studied for various flow regimes. The obtained results are as follows:

  • (1)

    The constitutive equations that specify the distribution parameter in various flow regimes have been discussed briefly.

  • (2)

    The constitutive equations that specify the relative motion between

Acknowledgements

The authors wish to thank Prof. J.Y. Lee (Handong Global University, South Korea) for his fruitful discussion. Part of this work was supported by the Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sport and Culture (no.: 14580542).

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