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

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

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

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

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

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

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:

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

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 plane. There are two important characteristics of this plane representation. First, when Co and are constant Eq. 3.15 shows that a linear relation exists between and ;*SPMlt*;*j*;*SPMgt*;. The slope of such a line gives the value of distribution parameter, Co, whereas the intercept with the axis gives the drift velocity, . 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 plane pertains to any abrupt measured changes in the value of Co and ; these indicate a change of flow-pattern in the conduct of the experiment.

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

This equation shows that can be determined if Co, and the average gas or vapor volumetric flux are known for a given flow regime; i.e., bubbly slug churn-turbulent. Noting that 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 and other flow-pattern dependent void fraction correlations are presented in Table 3.7 based on the work of these investigators.

<>P

**REFERENCES**

- C.J. Barozcy, "A Systematic Correlation for Two-Phase Pressure Drop,"
*Chem. Engr. Proj. Symp, Vol 62*, No 44, pp 232-249, 1966. -
A.E. Bergles, et al.,
*Two Phase Flow and Heat Transfer in the Power and Process Industries*, Hemisphere, New York, 1981. -
R. Bird, et al.,
*Transport Phenomena*, Wiley, New York, 1960. -
D. Chisholm, "The Pressure Gradient Due to Friction During the Flow of Boiling Water,"
*Engr & Boiler House Rev, Vol 73*, No 8, pp 252-256, 1966. -
J.G. Collier,
*Convective Boiling and Condensation*, McGraw-Hill, 2nd Edition, New York, 1981. -
L. Friedel, "Improved Friction Pressure Drop Correlations for Horizontal and Vertical Two-Phase Pipe Flow,"
*Proc European Two-Phase Group Mtg.*, JRC-Ispra, 1979. -
Y.Y. Hsu, H. Graham,
*Transport Processes in Two-Phase Boiling Systems*, Hemisphere, New York, 1977. -
M. Ishii,
*Thermo-Fluid Dynamic Theory of Two-Phase Flow*, Eyrolles, France, 1974. -
M. Ishii, "One-Dimensional Drift-Flux Model and Constitutive Equations for Relative Motion Between Phases in Various Two-Phase Flow Regimes,"
*Argonne National Lab Report, ANL 77-47*, October 1977. -
M. Ishii, N. Zuber, "Drag Coefficient and Relative Velocity in Bubbly, Droplet or Particulate Flows,"
*AIChE Jul, Vol 25*, No. 2, p 843, 1979. -
J.R.S. Thom, "Prediction of Pressure Drop During Forced Circulation of Boiling Water,"
*Intl Jul Heat Mass Transfer, Vol 7*, pp 709-724, 1964. -
G.B. Wallis,
*One-Dimensional Two-Phase Flow*, McGraw Hill, 2nd Edition, New York, 1979. -
N. Zuber, J. Findlay, "Average Volumetric Concentration in Two-Phase Systems,"
*Trans ASME Jul Ht Transfer, Vol 87*, p 453, 1969.

Last Modified: