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3.6. Void Fraction Prediction with the Drift Flux Model

As noted in the previous section, when the two phases are considered to have different velocities (e.g., liquid and gas), the relation between void fraction and quality is not analytically calculable, but requires some empirical data which links void and quality. A large number of empirical and semi-empirical methods have been suggested over the last fifty years. The semi-empirical model which seems to have the most physical basis is the drift flux model. It relates the gas-liquid velocity difference to the drift flux (or 'drift velocity') of the vapor relative to the liquid; e.g., due to buoyancy effects. This model has been principally developed by Zuber and Findlay (1965), Wallis (1969) and Ishii (1977), and has been refined since that time by themselves and coworkers. The model is fully developed in Wallis' book and only the essential relationships are presented here.

In all two-phase flows, the local velocity and local void fraction vary across the channel dimension, perpendicular to the direction of flow. To help us consider the case of a velocity and void fraction distribution (possibly different) it is convenient to define an average and void fraction weighted mean value of local velocity, v. Let F be parameters, such as any one of these local parameters, and an area average value of F across a channel cross-section would be given as

equation682

A void fraction weighted mean value of F may be defined as follows:

equation688

Now consider the gas velocity, tex2html_wrap_inline4597 , that can vary across the channel. An expression for the weighted mean gas velocity may be expressed as

equation695

Taking now a reference frame moving with the velocity of the center of the volume of the fluid one can define the gas or vapor phase drift velocity by

equation702

where j is the volumetric flux of the two-phase flow. This is the velocity of the center of volume of the mixture, and is given by

equation708

where tex2html_wrap_inline563 and tex2html_wrap_inline565 are the volumetric flux of the liquid and gas phases, respectively, given by

equation714

equation719

Using Eqs. 3.10 and 3.13 in Eq. 3.9, the weighted mean velocity of the gas or vapor phase can be expressed as:

equation724

equation738

where tex2html_wrap_inline4605 is the weighted mean drift velocity of the gas phase, and Co is the distribution parameter defined by

equation750

Therefore, Co depends on the form of the velocity and concentration (flow-pattern) profiles. For a given flow pattern, the extensive study of Zuber and Wallis suggest that Co depends on pressure, channel geometry and perhaps the flow rate.

The form of Eq. 3.15 suggests a plot of the data in tex2html_wrap_inline4607 plane. There are two important characteristics of this plane representation. First, when Co and tex2html_wrap_inline4609 are constant Eq. 3.15 shows that a linear relation exists between tex2html_wrap_inline4611 and ;SPMlt;j;SPMgt;. The slope of such a line gives the value of distribution parameter, Co, whereas the intercept with the tex2html_wrap_inline4611 axis gives the drift velocity, tex2html_wrap_inline4609 . Thus, these two parameters can be easily determined from experiments even when the measured profiles required for the direct calculation of Co are not available. The second important characteristic of the tex2html_wrap_inline4619 plane pertains to any abrupt measured changes in the value of Co and tex2html_wrap_inline4611 ; these indicate a change of flow-pattern in the conduct of the experiment.

The average void fraction tex2html_wrap_inline4623 at a given location in the flow channel can now be obtained by rearranging Eq. 3.15 giving:

equation770

This equation shows that tex2html_wrap_inline4623 can be determined if Co, tex2html_wrap_inline4627 and the average gas or vapor volumetric flux tex2html_wrap_inline4629 are known for a given flow regime; i.e., bubbly slug churn-turbulent. Noting that tex2html_wrap_inline4631 and Co are flow-pattern dependent quantities then any void fraction predictions based on Eq. 3.17 would reflect the flow-pattern effects on the void fraction.

Suggested expressions for Co and tex2html_wrap_inline4627 and other flow-pattern dependent void fraction correlations are presented in Table 3.7 based on the work of these investigators.

tabular798

tabular869

tabular938

tabular1053

<>P

tabular1144

tabular1194

tabular1245

REFERENCES


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Next: SOUND SPEED AND CRITICAL Up: MULTIPHASE FORMULATION AND PRESSURE Previous: Pressure Drop in a


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Last Modified: Mon Aug 4 00:56:50 CDT 1997