Two-phase friction multipliers

The Lockhart and Martinelli (1949) correlation also uses a two-phase friction multiplier, defined by Eq. (5.16). The friction multiplier has been correlated in terms of the Lockhart-Martinelli parameter, X, given by... 

Zhao and Bi (2001b) measured pressure drop in triangular conventional size channels d = 0.866—2.866 mm). The variations of the measured two-phase frictional multiplier with the Martinelli parameter X for the three miniature triangular channels used in experiments are displayed, respectively, in Fig. 5.29a-c. In Fig. 5.29 also shown are the curves predicted by Eq. (5.25) for C = 5 and C = 20. It is evident from Fig. 5.29 that the experimental data are reasonably predicted by the Lockhart-Martinelli correlation, reflected by the fact that all the data largely fall between the curves for C = 5 and C = 20, except for the case at very low superficial liquid velocities. 

Figure 5.31 shows a comparison of the two-phase friction multiplier data with the values predicted by Eq. (5.25) with C = 5, for both phases being laminar, and with C = 0.66 given by Mishima and Hibiki s (1996) correlation. It is clear that the data correlate well using a Lockhart-Martinelli parameter, but the predictions of... 

Fig. 5.29a-c Two-phase frictional multiplier 0 vs. Lockhart-Martinelli parameter X (Lockhart and Mar-tinelli 1949). Reprinted from Zhao and Bi (2001b) with permission... 

Fig. 5.31 Variation of two-phase friction multiplier data with Lockhart-Martinelli parameter. Reprinted from Kawahara et al. (2002) with permission...

To measure all the parameters pertinent to simulating reactor conditions, Ny-lund and co-workers (1968, 1969) presented data from tests carried out on a simulated full-scale, 36-rod bundle in the 8-MW loop FRIGG at ASEA, Vasteras, Sweden (Malnes and Boen, 1970). Their experimental results indicate that the two-phase friction multiplier in flow through bundles can be correlated by using Becker s correlation (Becker et al., 1962),... 

Figure 3.52 Mass flow modified coefficient in the Becker two-phase friction multiplier. (From Malnes and Boen, 1970. Copyright 1970 by Office for Official Publication of the European Community, Luxembourg. Reprinted with permission.)...

Souza et al. [157] measured two-phase pressure drops during turbulent flow of R-12 and R-134a and developed an expression for the two-phase frictional multiplier that successfully predicted their data to within 10 percent ... 

The separated flow model (for more details, see Collier and Thome [54]) considers that the phases are artificially segregated into two steams one liquid and one vapor, and has been continuously developed since 1949 when Lockhart and Martinelli [56] published their classic paper on two-phase gas-liquid flow. The main goal in this approach is to find an empirical correlation or simplified concept to relate the two-phase friction multiplier, ( ), to the independent variables of the flow. For example, the... 

The Armand model was chosen to calculate the two-phase friction multiplier, through Eq.2. 

Table 1 shows comparisons between pressure drop data for a full-scale channel assembly and the PATRIARCH code predictions. The latter have been based on empirically determined two-phase friction multipliers which seem to have a wide range of validity under SGHW conditions. In general, three two-phase pressure drop friction factors for rod clusters lie between the predictions of the Martlnelli-Nelson and Thom correlations for single round tubes. 

The equation for the mass flux effect, AF, has been obtained by correlating the measured friction multiplier values by means of regression analyses (Fig. 3.52). It is assumed that the two-phase friction loss in the channel is essentially unchanged by the presence of spacers. However, the increase in total pressure drop is determined by its presence in rod bundles (Janssen, 1962). 

That is, the two-phase frictional pressure gradient is calculated from a reference single-phase frictional pressure gradient (dP/dx)R by multiplying by the two-phase multiplier, the value of which is determined from empirical correlations. In equation 7.73 the two-phase multiplier is written as < >% to denote that it corresponds to the reference single-phase flow denoted by R. 

The value of the square root of the two-phase multiplier is read from Figure 7.13, or calculated from equation 7.85 or 7.86, and the two-phase frictional pressure gradient calculated from... 

Various studies have been conducted in predicting the two-phase frictional pressure losses in pipes. The Lockhart-Martinelli correlations [19] shown in Figure 3-7 are employed. The basis of the correlations is that the two-phase pressure drop is equal to the single-phase pressure drop of either phase multiplied by a factor that is derived from the singlephase pressure drops of the two-phases. The total pressure drop is based on the vapor-phase pressure drop. The pressure drop computation is based on the following assumptions ... 

The ratio of two-phase friction pressure loss to single-phase pressure loss was computed by integrating local pressure gradients at a constant pressure and heat flux from saturated liquid up to any quality x. Figure 22.16 illustrates two-phase friction pressure losses as a function of exit quality and system pressure. The friction pressure loss for steam-water mixtures at any pressure and exit quality, where quality varies linearly in the channel, is obtained by multiplying the friction loss for total flow rate as liquid by the ratio obtained from the appropriate curve in Figure 22.16. 

Probably the most widely used method for estimating the drop in pressure due to friction is that proposed by LOCKHART and Martinelli(,5) and later modified by Chisholm(,8 . This is based on the physical model of separated flow in which each phase is considered separately and then a combined effect formulated. The two-phase pressure drop due to friction — APtpf is taken as the pressure drop — AP/, or — APG that would arise for either phase flowing alone in the pipe at the stated rate, multiplied by some factor 2L or . This factor is presented as a function of the ratio of the individual single-phase pressure drops and ... 

The terms represent, respectively, the effect of pressure gradient, acceleration, line friction, and potential energy (static head). The effect of fittings, bends, entrance effects, etc., is included in the term Ke correlated as a number of effective velocity heads. The inclination angle 0 is the angle to the horizontal from the elevation of the pipe connection to the vessel to the discharge point. The term bi is the two-phase multiplier that corrects the liquid-phase friction pressure loss to a two-phase pressure loss. Equation (23-39) is written in units of pressure/density. 

In the separated flow models presented in Sections 7.7 and 7.8, the method of calculating the frictional component of the pressure gradient involves use of the two-phase multiplier 4>1 2 3 4 defined by... 

In contrast to the case of the homogeneous model, the accelerative term cannot be put in a simpler form because the phase velocities differ. It is therefore necessary to carry out the differentiation in the accelerative term. When this is done and the frictional component of the pressure gradient is represented using the wholly liquid two-phase multiplier, the resulting form of the momentum equation is... 

Lockhart and Martinelli (1949) used only liquid and only gas reference flows and, having derived equations for the frictional pressure gradient in the two-phase flow in terms of shape factors and equivalent diameters of the portions of the pipe through which the phases are assumed to flow, argued that the two-phase multipliers and 4>g could be uniquely correlated against the ratio X2 of the pressure gradients of the two reference flows ... 

A particularly simple and frequently used method comes from Lockhart and Martinelli [4.84], It is based on measurements of air-water and air-oil mixtures in horizontal tubes at low pressure. However the procedure has also proved itself in upward, vertical flow of two-phase single and multicomponent mixtures. The basic idea of the Lockhart-Martinelli method is that the frictional pressure drop in a two-phase flow can be determined, with use of a correction factor, from the frictional pressure drop in the individual phases. This means that the two-phase multipliers and are defined according to (4.127) and (4.128). 

TABLE 14.2 Two-Phase Flow Frictional Multiplier Correlations... 

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