Two Phase Flow. Lockhart-Martinelli calculation method

PRESSURE DROP FOR HORIZONTAL TWO PHASE FLOW

The pressure drop for systems of any type has three components that control the flow and loss of energy: friction, acceleration and lift. In the case of horizontal 2 phase flow. In this unit we will describe the 2 phase pressure drop excluding the elevation effect for horizontal lines; it will be detailed in next chapters.

Lockhart-Martinelli correlation.

This correlation indicates that the pressure drop in two phases flow can be calculated using the equations and graphs commonly used to calculate the pressure drop to a single fluid phase, once the individual velocities of each phase are known. By assuming that the two phases are running on the line totally separated from each other, it is possible to define their respective speeds in terms of a so-called hydraulic diameter and a shape factor. In their article, they publish in detail the analysis of the phenomenon and the development of this correlation.

Lockhart and Martinelli launched two basic postulates for their analysis:

- The static pressure drop for the liquid phase is always equivalent to the gas phase regardless of the flow pattern adopted by the moving mixture; also there is no appreciable radial static pressure difference.

- The volume occupied by the liquid phase plus the volume occupied by the gas phase at any moment and position, must be equal to the total volume of the pipe.

These postulates suggest the non-existence of a change in the flow pattern along the pipe. In this way, slug and plug flows patterns are eliminated in this consideration.

Based on their experimental observations, ?Lockhart and Martinelli plotted ? vs. X in a similar way to that shown in the attached figure. These researchers actually obtained four ? curves for each phase, as they defined the following flow regimes:

1.- Both liquid and gas phases in turbulent regime (tt).

2.- Turbulent flow in the liquid phase and viscous flow in the gas phase (tv).

3.- Viscous flow in the liquid and turbulent in the gas phase (vt).

4.- Both phases in viscous regime (vv).

Lockhart-Martinelli plot for ? and R as a function of X. (1949)

Lockhat Martinelli found that using a separate phase model, the general equation for frictional pressure losses in two-phase flow was given by:

?P2f ?=??^2??P1f

?P2f = pressure drop in two-phase flow.

?P1f = pressure drop in one of the phases, calculated by Darcy Wiesbach.

? ^ 2 = function that depends on the Lockhart-Martinelli modulus

X = 〖?Pl / ?Pg〗 ^ 0.5

X = Lockhart-Martinelli modulus for liquid-gas two-phase flow.

?Pl = pressure drop in the liquid phase.

?Pg = gas phase pressure drop?

Subsequently, other researchers undertook the task of developing more detailed and complex theoretical models, which provide the engineer with greater precision in the calculation of pressure drops in two-phase flow.

Lockhart-Martinelli calculation method:

1.- Determine the flow regime of each phase according to the Lockhart-Martinelli criterion, calculating ?Reynolds number for each phase:

Re = D V ρ / μ

D = internal diameter of the pipe in m.

V = superficial velocity of the phase in m / s.

ρ = density of the phase at operating conditions in kg / m3.

?μ = viscosity of the phase in kg / (m s).

?

If Re <1000, the phase regime is viscous (v).

If Re> 2000, the phase regime is turbulent (t).

This statement was defined by the researchers, different as was defined by Colebrook White.

2.- Calculate the pressure drop for each phase, using the Darcy Wiesbach equation:

h = f L V ^ 2 / (2 D g ρl)

f = Darcy's friction factor.

V = superficial velocity of the phase in m / s.

L = length of the pipe section in m.

ρl = density of the phase in kg / m3.

g = 9.81 m kg / (s2 kgf) = 32.2 ft lb / (s2 lbf)

D = internal diameter of the pipe in m..

Remember that the value of h is equivalent to the pressure drop of each phase.

To find the friction factor, use the Know approximations.

If Re <2100, the phase is in a laminar regime. The friction factor is only a function of the Reynolds number

??f = 64 / Re

If Re > 2100, the phase in question is in a transition regime. If?>?10,000, the phase flows at a turbulent rate. For these last two regimes, the Darcy friction factor is then a function of the relative roughness of the pipe (? / D) and the Reynolds number, and can be obtained using an approximation to the Moody plot, the Colebrook and White explicit equation, Haalan′s implicit equation that is gaining ground in the f calculation for fluids in turbulent regime (1984).

It is worth highlighting the difference between this transition criterion from laminar to turbulent regime (Re = 2100), and the Lockhart-Martinelli defined in step 1 of this method.

3.- Calculate the Lockhart Martinelli factor X with the equation

X = 〖?Pl / ?Pg〗 ^ 0.5

4.- The coefficients ?g and ?l are estimated using the equations developed by Turner and Walis (1965) for gas and Chilholm (1967) for liquids. These are not the original equations enunciated by Lockhart Martinelli, but subsequent research has accepted these correlations as the most appropriate.

Gas Phase

〖?g〗 ^ 2 = (〖1+ (x ^ (4 / (5-n)))〗 ^ █ ((5-n) / 2 @))

n = 1 viscous flow n = 0.75 for turbulent flow

Liquid Phase

〖?l〗 ^ 2 = 1 + c / x + 1 / x ^ 2

C = 20 turbulent liquid turbulent gas

C = 12 viscous liquid turbulent gas

C = 10 turbulent liquid viscous gas

C = 5 viscous liquid gas viscous

5.- The pressure drop of each phase is calculated using the Martinelli correlations

〖?g〗 ^ 2 * ?pgdarcy = ?Pg2f

〖?l〗 ^ 2 * ?pldarcy = ?Pl2f

The pressure drop of the section is the highest of the previous calculation.

Various studies have shown that the Lockhart and Martimelli equations, with the modifications made by other researchers, present deviations of up to 20% for pipes with a diameter greater than 6 inches, where the pressure drop values are greater than those observed in the field.

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It is important to indicate that studies carried out by Dukler focused on the analysis of non-separated phases, found correlations that reproduce the field values observed for pipes with diameters greater than 6 inches with deviations between 15 and 20%.? In a later article we will describe the Dukler method and the calculation for inclined and vertical pipes.

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Reference

-??????Jukka Kiijarvi

????????Darcy Friction Factor Formulae in Turbulent Pipe Flow-

????????Helsinki University of Technology.??Lunowa 2011 Fluid Mechanics Paper 110727

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-???????Saied Mokhatab, Wuillian Poe?Natural Gas Transmission and Processing

??????????2° edition Gulf Professional Publishing

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-??????Raja Venkaterwar ?A METHOD TO CALCULATE PRESSURE DROP FOR GAS-LIQUID FLOW IN LONG HORIZONTAL TRANSMISSION LINES Oklahoma State University 1987