ThermodynamicsのA-Zガイド、熱&アンプ、質量移動、流体工学

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Void fraction in a Gas-Liquid Flow may be defined as:

1. The fraction of the channel volume that is occupied by the gas phase.

2. The fraction of the channel cross-sectional area that is occupied by the gas phase.

Normally, in considering void fraction, we use the time-average value (taken over a long period of time), but it should be appreciated that void fraction is fluctuating with time and instantaneous values are also of interest. At a given point in the flow, the local fluid is either gas or one of the other phases. The probability of finding gas at a given point may be determined using local probes (see Void Fraction Measurement) and is referred to as the local void fraction. Void fraction is related to Slip Ratio S by the equations:

(1) where UG and UL are the superficial velocities of the gas and liquid phases, pL and pG the liquid and gas densities and x the flow quality (see article on Multiphase Flow for definitions). For a homogeneous flow, S = 1 and:

(2) where β is the volume flow fraction of the gas phase.

Usually, however, S ≠ 1 and one must produce a model which includes differences in velocity between the phases. A variety of such models have been derived, ranging from simple one-dimensional models, to empirical correlations and to more complex phenomenological models. Each of these approaches will now be briefly discussed.

### One-dimensional flow method

Here, it is assumed that the velocity of the phases and the void fraction are constant across the channel cross-section. A drift flux jGL is defined as the gas flux relative to a plane moving along the channel at the total superficial velocity U (= UL + UG). Determination of the void fraction involves simultaneous solution of the continuity relationship:

(3) and a relationship describing the physics of the system:

(4) This one-dimensional flow model is used principally for bubble flow systems where the physical relationship is often written in the form:

(5) where u∞ is the rise velocity of a single bubble in a static pool of the liquid phase, and n is an exponent whose value is typically in the range 1.5-2.0. The simultaneous solution of Eqs. (3) and (5) is illustrated graphically in Figure 1. Solutions are possible for both cocurrent upward flow and cocurrent downward flow (though the void fraction is considerably higher in the latter case). For the gas flowing upwards and the liquid flowing down, no solutions are possible above a given liquid velocity. Below this, two solutions are possible as shown. Figure 1. Solutions for void fraction in vertical flow, using the one-dimensional analysis method, (a) Vertical cocurrent up flow, (b) Vertical cocurrent down flow, (c) Vertical countercurrent flow (liquid down, gas up), (d) Vertical countercurrent flow (liquid up, gas down).

For countercurrent flow with the gas going down and the liquid going up, no solutions are possible, of course, as is also shown.

Further details of the one-dimensional model are given by Wallis (1969) and Hewitt (1982).

The one-dimensional flow model can be developed to take account of variations of velocity and void fraction across the channel, and this leads to the Drift Flux Model, which is described in a separate article.

### Empirical correlations

A vast range of empirical correlations have been developed for void fraction, and it is beyond the scope of the present article to deal with these in detail. One of the earliest correlations (still widely used) is that of Lockhart and Martinelli (1949) who correlated εG as a function of the Martinelli parameter X which is defined as:

(6) where (dpF/dz)L and (dpp/dp)G are the pressure gradients for the single phase flow of the liquid phase and the gas phase,respectively, flowing alone in the channel. The correlation was in graphical form, and is illustrated in Figure 2. Figure 2. Lockhart and Martinelli (1949) correlation for void fraction.

The Lockhart-Martinelli correlation gives an inadequate representation of the effect of mass flux on void fraction. Gross errors can occur, and this has led to the development of a wide variety of alternative correlations. One of the best known of these is the so-called CISE correlation (developed at the CISE laboratories in Milan) which is presented by Premoli et al. (1971). The correlation is given in the form of slip ratio S (which can be related to void fraction using Eq. (1) above). The correlation has the form:

(7) where

(8) where β is the volume flow fraction (Eq. (2)). E1 and E2 are given by:

(9) (10) where:

(11) (12) where is the mass flux, D the diameter of the tube, ηL the viscosity of the liquid and σ the surface tension.

As will be seen from the above, a large number of empirical factors have been introduced into this correlation in order to make it fit the data available at the time of its preparation. Data is constantly being generated and the empirical correlations become ever more complex as they are adjusted to cope with the widening data base. A more recent correlation, written in terms of the drift flux parameters (see article on Drift Flux Models) is that of Chexal and Lellouche (1991), and this contains over twenty arbitrary constants! Nevertheless, it is a very good representation of the data base, which justifies its wide utilization.

### Phenomenological models for void fraction

In an attempt to avoid the arbitrary nature of empirical correlations, many investigators have attempted to develop phenomenological models for void fraction and other parameters in two-phase flows. The first step is to identify the flow pattern or flow regime (see entry on Gas-Liquid Flow). Having identified the flow regime, it is then possible to construct a detailed model for the given flow configuration. Examples of such phenomenological modeling are given in the entries on Plug Flow, Annular Flow, Bubble Flow, Stratified Flow, Slug Flow and Churn Flow.

#### REFERENCES

Chexal, B. and Lellouche, G. (1991) Void fraction correlation for generalized applications, Nuclear Safety Analysis Centre of the Electric Power Research Institute, Report NSAC/139.

Hewitt, G. F. (1982) Void fraction, Handbook of Multiphase Systems, Ch, 2.3 G. Hetsroni, Ed., McGraw-Hill, New York. ISBN 0-07-028460-1.

Lockhart, R. W. and Martinelli, R. C. (1949) Proposed correlation of data for isothermal two-phase, two-component flow in pipes, Chem. Eng. Prog., 45, 39-48.

Premoli, A., Francesco, D. and Prina, A. (1971) A dimensionless correlation for determining the density of two-phase mixtures, Termotecnica, 25, 17-26,

Wallis, G. B. (1969) One-Dimensional Two-Phase Flow, McGraw-Hill, New York.

#### References

1. Chexal, B. and Lellouche, G. (1991) Void fraction correlation for generalized applications, Nuclear Safety Analysis Centre of the Electric Power Research Institute, Report NSAC/139.
2. Hewitt, G. F. (1982) Void fraction, Handbook of Multiphase Systems, Ch, 2.3 G. Hetsroni, Ed., McGraw-Hill, New York. ISBN 0-07-028460-1.
3. Lockhart, R. W. and Martinelli, R. C. (1949) Proposed correlation of data for isothermal two-phase, two-component flow in pipes, Chem. Eng. Prog., 45, 39-48.
4. Premoli, A., Francesco, D. and Prina, A. (1971) A dimensionless correlation for determining the density of two-phase mixtures, Termotecnica, 25, 17-26,
5. Wallis, G. B. (1969) One-Dimensional Two-Phase Flow, McGraw-Hill, New York.