Lockhart parameter

Rapid approximate predictions of pressure drop for fully developed, incompressible horizontal gas/fiquid flow may be made using the method of Lockhart and MartineUi (Chem. Eng. Prog., 45, 39 8 [1949]). First, the pressure drops that would be expected for each of the two phases as if flowing alone in single-phase flow are calculated. The LocKhart-Martinelli parameter X is defined in terms of the ratio of these pressure drops ... 

For fully developed incompressible horizontal gas/hquid flow, a quick estimate for Ri may be obtained from Fig. 6-27, as a function of the Lockhart-MartineUi parameter X defined by Eq. (6-131). Indications are that liquid volume fractious may be overpredicled for liquids more viscous than water (Alves, Chem. Eng. Prog., 50, 449-4.56 [19.54]), and uuderpredicted for pipes larger than 25 mm diameter (Baker, Oil Gas]., 53[12], 185-190, 192-195 [1954]). 

This can be readily integrated numerically as long as we use the appropriate nonequilibrium equivalent specific volume in the integration. A reasonably simple form for has been suggested by Chisholm (1983), which makes use of estabhshed correlations for the slip velocity K, which depends on the Lockhart-Martinelh parameter X. Integrating Eq. (26-118) gives ... 

For our purposes, a rough estimate for general two-phase situations can be achieved with the Lockhart and Martinelli correlation. Perry s has a writeup on this correlation. To apply the method, each phase s pressure drop is calculated as though it alone was in the line. Then the following parameter is calculated ... 

Lockhart and MaRTINELLI051 expressed hold-up in terms of a parameter X. characteristic of the relative flowrates of liquid and gas, defined as ... 

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...

Lockhart-Martinelli parameter Volumetric quality, void fraction Streamwise coordinate... 

This matter was discussed in Sect. 5.8. For channeis of dh = 0.9-3.2 mm, the two-phase pressure drop can be caicuiated using the Lockhart-Martineiii modei with parameter C, ranging from 5 to 20. The parameter C decreases when the hydraulic diameter decreases (Zhao and Bi 2001). For channels of = 100 pm, (Kawahara et al. 2002) two-phase pressure drop can be correlated within an accuracy of 10% using the Lockhart-Martineiii model with C = 0.24. 

The two-phase correction factor fc is obtained from Figure 12.56 in which the term l/Xtt is the Lockhart-Martinelli two-phase flow parameter with turbulent flow in both phases (See Volume 1, Chapter 5). This parameter is given by ... 

A separated flow model for stratified flow was presented by Taitel and Dukler (1976a). They indicated analytically that the liquid holdup, R, and the dimensionless pressure drop, 4>G, can be calculated as unique f unctions of the Lockhart-Martinelli parameter, X (Lockhart and Martinelli, 1949). Considering equilibrium stratified flow (Fig. 3.37), the momentum balance equations for each phase are... 

Here 7 was shown to be essentially independent of the Lockhart-Martinelli parameter, X, for values of (1/30 greater than unity. Further study, however, is necessary to develop a generalized equation for the coefficient 7. 

When both phases are in turbulent flow, or when one phase is discontinuous as in bubble flow, it is not presently possible to formulate the proper boundary conditions and to solve the equations of motion. Therefore, numerous experimental studies have been conducted where the holdups and/or the pressure drop were measured and then correlated as a function of the operating conditions and system parameters. One of the most widely used correlations is that of Lockhart and Martinelli (L12), who assumed that the pressure drop in each phase could be calculated from the equations... 

Lockhart and Martinelli divided gas-liquid flows into four cases (1) laminar gas-laminar liquid (2) turbulent gas-laminar liquid (3) laminar gas-turbulent liquid and (4) turbulent gas-turbulent liquid. They measured two-phase pressure drops and correlated the value of 0g with parameter % for each case. The authors presented a plot of acceleration effects, incompressible flow (3) no interaction at the interface and (4) the pressure drop in the gas phase equals the pressure drop in the liquid phase. 

Martinelli and Nelson (M7) developed a procedure for calculating the pressure drop in tubular systems with forced-circulation boiling. The procedure, which includes the accelerative effects due to phase change while assuming each phase is an incompressible fluid, is an extrapolation of the Lockhart and Martinelli x parameter correlation. Other pressure drop calculation procedures have been proposed for forced-circulation phase-change systems however, these suffer severe shortcomings, and have not proved more accurate than the Martinelli and Nelson method. 

Void fraction and square root of two-phase multiplier against Martinelli parameter X Source R. W. Lockhart and R. C. Martinelli, Chemical Engineering Progress 45, pp. 39-46(1949)... 

The Lockhart-Martinelli correlation provides the relationship between 4>j. and the Martinelli parameter X . Therefore, use of equation 7.95 enables the relationship between 4>lo and X at low pressures to be found. 

Dengler and Addoms 8 measured heat transfer to water boiling in a 6 m tube and found that the heat flux increased steadily up the tube as the percentage of vapour increased, as shown in Figure 14.4. Where convection was predominant, the data were correlated using the ratio of the observed two-phase heat transfer coefficient (htp) to that which would be obtained had the same total mass flow been all liquid (hi) as the ordinate. As discussed in Volume 6, Chapter 12, this ratio was plotted against the reciprocal of Xtt, the parameter for two-phase turbulent flow developed by Lockhart and Martinelli(9). The liquid coefficient hL is given by ... 

Davis (Dl) has suggested that the introduction of the Froude number into the Lockhart-Martinelli parameter. A, gives a description of gravitational and inertial forces so that this model can be applied to vertical flow. The revised parameter, X, is defined empirically for turbulent-turbulent flow as,... 

Hughmark and Pressburg (H12) have correlated statistically their void data and others for vertical flow, using a modified Lockhart-Martinelli parameter, X, given as... 

Entrainment studies have been relatively few, as pointed out earlier. Anderson and Mantzouranis (A3) used the results of measurements of entrainment (which was small in their work) to correct their calculated liquid film thickness, and thus obtained somewhat better agreement with experimental values. Wicks and Dukler (W2) measured entrainment in horizontal flow, and obtained a correlation for the amount of entrainment in terms of the Lockhart and Martinelli parameter, X. The entrainment parameter, R, of Wicks and Dukler is given by... 

TJse of the Lockhart-Martinelli Parameters for Heat Transfer... 

For sufficiently large heat flux to mass flow ratios, the nucleation mechanism predominates and the heat transfer becomes independent of the two-phase flow characteristics of the system. Thus at large values of the boiling number, the heat transfer coefficients are virtually independent of the Lockhart-Martinelli parameter, Xn. 

Again referring to Fig. 13, the same general trend is apparent in both the pressure-drop and number-of-transfer-unit curves. This suggests that another empirical correlating procedure could be arrived at for example, an approximate relationship exists between the length of a transfer unit (LTU) and the Lockhart-Martinelli parameters, X. 

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