Two phase frictional

The basis for single-phase and some two-phase friction loss (pressure drop) for fluid flow follows the Darcy and Fanning concepts. The exact transition from laminar or dscous flow to the turbulent condition is variously identified as between a Reynolds number of 2000 and 4000. 

The two-phase frictional pressure gradient is obtained from ... 

Several models have been proposed to evaluate the two-phase mixture viscosity, and the model selected may affect the predicted two-phase frictional pressure drop ... 

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. 5.30 Comparison of the two-phase frictional pressure gradient between micro-channel data and homogeneous flow model predictions using different viscosity formulations. Reprinted from Kawahara et al. (2002) with permission...

Finally, a comparison of the two-phase frictional pressure gradient data with the predictions of the Lockhart-Martinelli correlation using different C-values is shown in Fig. 5.32, including C = 5, C = 0.66, C calculated from the Lee and Lee model (2001), and C = 0.24. The conventional value of C = 5 again significantly over-... 

In Table 6.7, C is the Martinelli-Chisholm constant, / is the friction factor, /f is the friction factor based on local liquid flow rate, / is the friction factor based on total flow rate as a liquid, G is the mass velocity in the micro-channel, L is the length of micro-channel, P is the pressure, AP is the pressure drop, Ptp,a is the acceleration component of two-phase pressure drop, APtp f is the frictional component of two-phase pressure drop, v is the specific volume, JCe is the thermodynamic equilibrium quality, Xvt is the Martinelli parameter based on laminar liquid-turbulent vapor flow, Xvv is the Martinelli parameter based on laminar liquid-laminar vapor flow, a is the void fraction, ji is the viscosity, p is the density, is the two-phase frictional... 

It was shown that a normalized version of the two-phase friction factor, CfTpl CfQ, is uniquely related to X. The normalizing friction factor, CfQ, is calculated from single-phase friction factor correlations using a Reynolds number calculated as if both phases flow as liquid,... 

Figure 3.42 Two-phase friction pressure drop correlation for G = 1 x 106 lb/hr ft2. (From Baroczy. 1966. Copyright 1966 by Rockwell International, Canoga Park, CA. Reprinted with permission.)...

Figure 3.48 Comparison of potassium and sodium two-phase friction pressure drop data with Lock-hart-Martinelli correlation, and with a simple correlation [1/(1 - a)]. (From Fauskeand Grolmes, 1970. Copyright 1970 by American Society of Mechanical Engineers. Reprinted 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),... 

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). 

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.)...

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. 

When the reference flow is the whole of the two-phase flow as liquid, then the two-phase frictional pressure gradient is given by... 

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... 

In summary, the calculation of pressure drops by the Lockhart-Marti-nelli method appears to be reasonably useful only for the turbulent-turbulent regions. Although it can be applied to all flow patterns, accuracy of prediction will be poor for other cases. Perhaps it is best considered as a partial correlation which requires modification in individual cases to achieve good accuracy. Certainly there seems to be no clear reason why there should be a simple general relationship between the two-phase frictional pressure-drop and fictitious single-phase drops. As already pointed out, at the same value of X in the same system, it is possible to have two different flow patterns with two-phase pressure-drops which differ by over 100%. The Loekhart-Martinelli correlation is a rather gross smoothing of the actual relationships. 

By analogy to single-phase flow, the two-phase frictional pressure-drop can be expressed by the conventional Fanning equation, and thereby a friction factor is defined. These friction factors may be based on liquid properties, gas properties, or on a fictitious single fluid of mean properties obtained by some averaging procedure. Typical definitions, such as those shown in equation (32), have been given and discussed recently by Govier and Omer (G4) ... 

The definition of friction factor using mean fluid properties has been most widely used because it reduces to the correct single-phase value for both pure liquid and pure gas flow. This technique is very similar to the so-called homogeneous model, because it has a clear physical significance only if the gas and liquid have equal velocities, i.e., without slip. Variations of this approach have also been used, particularly the plotting of a ratio of a two-phase friction factor to a single-phase factor against other variables. This approach is then very similar to the Lockhart-Martinelli method, since it can be seen that (G4)... 

Attempts have been made to compare the experimentally measured pressure drops with various of the theories discussed in Section III, F and with the more generalized empirical two-phase friction-factor correlations of the type discussed by Dukler and Wicks (D17). Hewitt et al. (H10) have compared their experimental data for upward cocurrent flow with one such empirical correlation and with the theories of Anderson and Mantzouranis (A5) and of Dukler (D12) and Hewitt (H7). In most cases there was good qualitative agreement only. 

Step 2. The ratio of two-phase friction factor to the gas-phase friction factor in the pipeline is determined here. 

Figure 6.8. Friction factors and void fractions in flow of single phase fluids in granular beds, (a) Correlation of the two-phase friction factor. Re = DpG/( — s)p and fp = gcDp /pu — a)] P L) = 150/Re + 4.2/(Re) [Sato et al, J. Chem. Eng. Jpn. 6,147-152 1973)]. (b) Void fraction in granular beds as a function of the ratio of particle and tube diameters. [Lem, Weintraub, Grummer, Pollchik, and Storch, U.S. Bur. Mines Bull. 504 1951)].

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