## X2

Table 15-2 Values of Constant C in Two-Phase Multiplier Equations

Flow state Liquid Gas C

tt Turbulent Turbulent 20

vt Laminar Turbulent 12

tv Turbulent Laminar 10

vv Laminar Laminar 5

where the values of C for the various flow combinations are shown in Table 15-2.

The Lockhart-Martinelli correlating parameter x2 is defined as

9X/ fGm PGD

Here, /Lm is the tube friction factor based on the "liquid-only" Reynolds number NReLm = (1 — x)GmD/^L and/Gm is the friction factor based on the "gas-only" Reynolds number NReGm = xGmD/^G. The curves cross at X = 1, and it is best to use the "G" reference curves for x < 1 and the "L" curves for x > 1.

Using similarity analysis, Duckler et al. (1964b) deduced that

which is equivalent to the Martinelli parameters

and ' and pm are the equilibrium (no-slip) properties. Another major difference is that Duckler et al. deduced that the friction factors fL and fG should both be evaluated at the mixture Reynolds number,

5. Slip and Holdup

A major complication, especially for separated flows, arises from the effect of slip. Slip occurs because the less dense and less viscous phase exhibits a lower resistance to flow, as well as expansion and acceleration of the gas phase as the pressure drops. The result is an increase in the local holdup of the more dense phase within the pipe ('m) (or the corresponding two-phase density, pm), as given by Eq. (15-11). A large number of expressions and correlations for the holdup or (equivalent) slip ratio have appeared in the literature, and the one deduced by Lockhart and Martinelli is shown in Fig. 15-7. Many of these slip models can be summarized in terms of a general equation of the form

for which the values of the parameters are shown in Table 15-3. Although many additional slip models have been proposed in the literature, it is not clear which of these should be used under a given set of circumstances. In some cases, a constant slip ratio (S) may give satisfactory results. For example, in a comparison of calculated and experimental mass flux data for high velocity air-water flows through nozzles, Jamerson and Fisher (1999) found that S = 1.1-1.8 accurately represents the data over a range of x = 0.02-0.2, with the value of S increasing as the quality (x) increases.

A general correlation of slip is given by Butterworth and Hewitt (1977):

where

 Model ao a1 a2 a3 Homogeneous
0 0