Pressure Lockhart-Martinelli correlation

The two-phase pressure drop is obtained by multiplying either the liquid-phase drop by (t) or the gas-phase pressure drop by. Figure 7-23 gives the Lockhart-Martinelli correlation between X and ([t s... 

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. 

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-... 

Fig. 5.32 Predictions of two-phase friction pressure gradient data by a Lockhart-Martinelli correlation with different C-values. Reprinted from Kawahara et al. (2002) with permission...

This correlation is an extension of the Lockhart-Martinelli correlation. The earlier correlation is limited to low pressures and systems in which no change of phase occurs. Although Lockhart and Martinelli provided for four flow regimes, it is unusual in industrial processes for either phase to be in laminar flow. The Martinelli-Nelson (1948) correlation is specifically for forced circulation boiling of water in which it is assumed that both phases are in turbulent flow. 

When a change of phase occurs, as in boiling, it is necessary to use the wholly liquid reference flow (an only liquid basis would change as the liquid flow rate decreases during boiling). At low pressures, the results of the Lockhart-Martinelli correlation can be used for the frictional component of the pressure gradient but it is necessary to convert the only liquid basis used in the earlier correlation to the wholly liquid basis. It is assumed that the frictional pressure gradients for the two reference flows are related by the expression... 

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. 

It should be remembered that these correlations as originally devised by Lockhart and Martinelli were based almost entirely on experimental data obtained for situations in which accelerative effects were minor quantities. The Lockhart-Martinelli correlation thus implies the assumption that the static pressure-drop is equal to the frictional pressure-drop, and that these are equal in each phase. The Martinelli-Nelson approach supposes that the sum of the frictional and accelerational pressure-drops equals the static pressure-drop (hydrostatic head being allowed for) and that the static pressure-drop is the same in both phases. When acceleration pressure losses become important (e.g., as critical flow is approached), they are likely to be significantly different in the gas and liquid phases, and hence the frictional pressure losses will not be the same in each phase. In these circumstances, the correlation must begin to show deviations from experiment. 

There are a number of pressure drop correlations for two-phase flow in packed beds originating from the Lockhart-Martinelli correlation for two-phase flow in pipes. These correlate the two-phase pressure drop to the single-phase pressure drops of the gas and the liquid obtained from the Ergun equation. See, for instance, the Larkins correlation [Larkins, White, and Jeffrey, AIChE J. 7 231 (1967)]... 

Eqs. (Ga) and (Gb) for hokhq), /L had been derived. semi-empirically [6 J, and are given for the sake of completeness. In the range 0.01 predict values of cp that deviate from the values given by the Lockhart-Martinelli curve with an averaged absolute deviation of less than 4% and a maximum absolute deviation of 10%. Since the pressure drop is proportional to (p , the errors for pressure drop are slightly more than double these values. Such deviations are apt to be acceptable since the Lockhart-Martinelli correlations may sometimes give predictions with errors in excess of 50% of experimental data. 

These correlations require both pressure-drop and gas-holdup values which may be predicted by the Lockhart-Martinelli correlation (L22) or the Hughmark correlation (H13). 

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 ... 

In Table 17.22, two correlations are presented for shellside two-phase flow pressure drop estimation, based on modifications of the internal flow correlations. The first correlation uses the modified Chrisholm correlation [69, 79], and the second one [80] employs the modified Lockhart-Martinelli correlation. The first correlation is for horizontal crossflow (crossflow in a baffled horizontal heat exchanger with horizontal or vertical baffle cuts). The second one is for vertical crossflow (upflow in a horizontal tube bundle). 

Lazarek and Black [67] obtained good predictions of their data by using a value of 30 in the generalized Chisholm/Lockhart-Martinelli correlation for C. Mishima and Hibiki [66] obtained reasonably good predictions for their frictional pressure drop data for air-water flows by correlating the Chrisholm C parameter in the Lockhart-Martinelli correlation as a function of the tube diameter as follows ... 

Two correlations are widely used the Lockhart-Martinelli correlation and the homogeneous model of Dukler and others. Dukler s correlation will be slightly better for most applications. Both correlations are expected to give better accuracy for horizontal flow than for vertical flow. Only when the two-phase pressure drop is high will the estimate be relatively independent of pipe orientation. 

To estimate the two-phase pressure drop with the Lockhart-Martinelli correlation, the procedure outlined below is followed ... 

Figure 11. Friction pressure drop calculated with the Lockhart-Martinelli correlation using both liquid and two-phase equivalent length as a function of experimental friction pressure drop.

In this case there is good agreement between the values calculated for total pressure drop. Since gas flow is predominant, the total empty pipe pressure drop calculated from the gas-only pressure drop using eq. (7-41b) should be used. It should be noted that this Lockhart-Martinelli correlation is considered to be conservative when used in vertical downward flow. The original work was all in horizontal flow. 

The Lockhart-Martinelli correlations, as noted earlier, were developed for estimating the pressure drops associated with two-phase flow at ambient conditions. However, because of the increased vaporization tendency of cryogenic fluids, these correlations have a tendency to underestimate the pressure drop by as much as 10-30% with low-temperature flows. 

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 ... 

For a micro-channel connected to a 100 pm T-junction the Lockhart-Martinelli model correlated well with the data, however, different C-values were needed to correlate well with all the data for the conventional size channels. In contrast, when the 100 pm micro-channel was connected to a reducing inlet section, the data could be fit by a single value of C = 0.24, and no mass velocity effect could be observed. When the T-junction diameter was increased to 500 pm, the best-fit C-value for the 100 pm micro-channel again dropped to a value of 0.24. Thus, as in the void fraction data, the friction pressure drop data in micro-channels and conventional size channels are similar, but for micro-channels, significantly different data can be obtained depending on the inlet geometry. 

The Lockhart-Martinelli model can correlate the data obtained from pressure drop measurements in gas-liquid flow in channels with hydraulic diameter of 0.100-1.67 mm. The friction multiplier is 0l = 1 + C/X - -1 /X. ... 

Lockhart and Martinelli (1949) suggested an empirical void fraction correlation for annular flow based mostly on horizontal, adiabatic, two-component flow data at low pressures, Martinelli and Nelson (1948) extended the empirical correlation to steam-water mixtures at various pressures as shown in Figure 3.27. The details of the correlation technique are given in Chapter 4. Hewitt et al. (1962) derived the following expression to fit the Lockhart-Martinelli curve ... 

The model is a significant improvement over the Lockhart and Martinelli correlations for pressure drop and holdup (discussed in Sec. 3.5.3). A severe limitation of the model, however, is the dependency on the empirical expression for Cf j [Eq. 3-126]. This expression is based on air-water data only, and has not been shown to apply to other systems. 

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. 

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. 

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. 

A method for predicting pressure drop and volume fraction for non-Newtonian fluids in annular flow has been proposed by Eisen-berg and Weinberger (AlChE J., 25, 240-245 [1979]). Das, Biswas, and Matra (Can. J. Chem. Eng., 70, 431—437 [1993]) studied holdup in both horizontal and vertical gas/liquid flow with non-Newtonian liquids. Farooqi and Richardson Trans. Inst. Chem. Engrs., 60, 292-305, 323-333 [1982]) developed correlations for holdup and pressure drop for gas/non-Newtonian liquid horizontal flow. They used a modified Lockhart-Martinelli parameter for non-Newtonian... 

The pressure drop data in highly-pulsed- and spray-flow regimes were obtained in a 10.2-cm-i.d. clear acrylic column by the Pittsburgh Energy Research Center (PERC).22 Over 300 pressure-drop data points were obtained for both 0.635-cm x 0.635-cm and 0.32-cm x 0.32-cm pellets. Some of these data are shown in Fig. 7-5. These data were well correlated by Tallmadge s correlation,36 as shown in Fig. 7-6. Sato et al.,27 on the other hand, correlated their data with the Lockhart-Martinelli type of relation. They also graphically illustrated some pressure-drop data in all three bubble-, pulsed-, and spray- (or gas-continuous) flow regimes. 

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