Lockhart and Martinelli

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 fog or spray type flow, Ludwig cites Baker s suggestion of multiplying Lockhart and Martinelli by two. 

Lockhart and Martinelli used pipes of one inch or less in diameter in their test work, achieving an accuracy of about -l-/-50%. Predictions are on the high side for certain two-phase flow regimes and low for others. The same -l-/-50% accuracy will hold up to about four inches in diameter. Other investigators have studied pipes to ten inches in diameter and specific systems however, no better, generalized correlation has been found.The way... 

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 ... 

Empirical void fraction correlations, relating to the Lockhart-Martinelli factor X and from there to volumetric quality x, have been provided by Lockhart and Mar-tinelli (1949), Baroczy (1963), Wallis (1969), and have been discussed by Butter-worth (1975) and Chen and Spedding (1983). Butterworth (1975) showed that Lockhart and Martinelli s correlation (1949) for void fraction, as well as several other void fraction correlations, can be represented in the following generic form ... 

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... 

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 ... 

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... 

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. 

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. 

Numerous other correlations for pressure drop and holdup have been developed, but none has been accepted by the practicing engineer to the extent that Lockhart and Martinelli s has. Charles and Lilliheht (C2) have developed a correlation, which is analogous to that of Lockhart and Martinelli, for pressure drop in stratified laminar liquid-turbulent liquid systems. Unfortunately they did not include a holdup correlation. Anderson and Russell (A4) and Dukler et al. (D4) have reviewed the applicability and accuracy of the more useful correlations. A designer must be aware that, while a correlation is supposedly applicable to a specific flow pattern, it can yield greatly inaccurate results in some cases. 

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. 

For gas-liquid flows in Regime I, the Lockhart and Martinelli analysis described in Section I,B can be used to calculate the pressure drop, phase holdups, hydraulic diameters, and phase Reynolds numbers. Once these quantities are known, the liquid phase may be treated as a single-phase fluid flowing in an open channel, and the liquid-phase wall heat-transfer coefficient and Peclet number may be calculated in the same manner as in Section lI,B,l,a. The gas-phase Reynolds number is always larger than the liquid-phase Reynolds number, and it is probable that the gas phase is well mixed at any axial position therefore, Pei is assumed to be infinite. The dimensionless group M is easily evaluated from the operating conditions and physical properties. 

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 (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... 

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 ... 

In order to determine whether each phase is in laminar or turbulent flow, Lockhart and Martinelli suggested tentatively that the Reynolds number for the appropriate reference flow should be greater than 2000 for turbulent flow and less than 1000 for laminar flow. At intermediate values the flow was thought to be transitional. 

In developing their correlation, Lockhart and Martinelli assumed that the friction factors could be determined from equations of the same form as the Blasius equation ... 

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. 

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 ... 

In the above terms, the quantities (AP/AZ)i or aP/aZ)g are calculated from conventional single-phase correlations on the basis that the liquid or gas is flowing in the pipe alone at the same individual mass-flow rate as in the two-phase case. Lockhart and Martinelli (L6) have given the appropriate expressions for X, and the relationships between tpa, flow regimes. The relationships are shown graphically in Fig. 7. 

Fig. 7. Pressure drop correlation of Lockhart and Martinelli for frictional pressure losses in horizontal cocurrent flow.

Since 1949 the correlations of Lockhart and Martinelli, and of Martinelli and Nelson, have been used or checked by many investigators of two-phase flow phenomena. The assumptions made in the derivations of these correlations would seem to rule out certain flow patterns, e.g.. 

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. 

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... 

In the forced-convection region, Dengler and Addoms and also Guerrieri and Talty correlate their results by analogy to the Lockhart and Martinelli approach, and show that the following equations can be used ... 

In the original work on two-phase pressure-drops at the University of California (B13), tests were made in tubes inclined at 1 6, as well as in horizontal tubes. In general, these data for small inclinations also fitted the correlations proposed finally by Lockhart and Martinelli for horizontal tubes. 

Dimensionless correlations based on momentum-or energy balances, and using the Lockhart and Martinelli parameter, Xu that is the ratio of the single-phase pressure drops at the same velocities ... 

Dimensionless correlations based on momentum-or energy-balances and using the Lockhart and Martinelli parameter, Xc, or some similar parameter. The practical relevance of such correlations is limited since their use requires the knowledge of the single-phase pressure drop of both gas and liquid furthermore, the influence of the geometry of the bed is not always well described by these single-phase pressure drops alone. 

Some correlations of multipliers are listed in Table 6.8. Lockhart and Martinelli distinguish between the various combinations of turbulent and laminar (viscous) flows of the individual phases in this work the transition Reynolds number is taken as... 

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