Pressure drop in two-phase flow is a major design variable, governing the pumping power required to transport two-phase fluids and also governing the recirculation rate in natural circulation systems. The conservation equations for two-phase flow are a subset of those for Multiphase Flow. Full derivation of the conservation equations is given in the article Conservation Equations, Two-Phase. For the purposes of this article, we shall use simplified forms of the momentum equations which apply to ducts of constant cross section and for steady state flow. Extensive background information on the subject of pressure drop is given by Hewitt (1982) and a review of pressure drop in orifices, valves, bends and fittings is given by Hewitt (1984).

The simplest approach to the prediction of two-phase flows is to assume that the phases are thoroughly mixed and can be treated as a single-phase flow. This *homogeneous model* (see Multiphase Flow) will obviously work best when the phases are strongly interdispersed (i.e. at high velocities). For the homogeneous model, the pressure gradient (dp/dz) is given by:

where τ_{0} is the wall shear stress, P the tube periphery, S the tube cross sectional area, the mass flux, z the axial distance, g the acceleration due to gravity, α the angle of inclination of the channel to the horizontal and ρ_{H} the homogeneous density given by:

where ρ_{G} and ρ_{L} are the gas and liquid densities and x is the quality (fraction of the total mass flow which is vapour).

The three terms on the right-hand side of Eq. (1) may be regarded respectively as the frictional pressure gradient (−dp_{F}/dz), the acceleration pressure gradient (−pd_{a}/dz) and the gravitational pressure gradient (−dp_{g}/dz). Thus

The frictional pressure gradient term in the homogeneous model is often related to a two-phase friction factor f_{TP} as follows:

where D is the tube diameter. f_{TP} may be related (via the normal single-phase friction factor relationships) to a two-phase Reynolds number defined as follows:

where η_{TP} is a two-phase viscosity. There are some difficulties in defining the latter; a whole variety of forms being suggested in the literature. These are exemplified by the relationship:

where η_{G}and η_{L} are the gas and liquid viscosities respectively. In fact, though, the homogeneous model departs grossly from experimental data and simply readjusting the definition of viscosity has been found to be totally inadequate in bring agreement. Many authors have suggested empirical modifications of the friction factor to take account of the two-phase nature of the flow. Perhaps the most widely used of these corrections to the homogeneous model is the correlation of Beggs and Brill (1973), which corrects the homogeneous model for both flow regime and tube inclination. However, the preferred option has been to work using the
*separated flow model*, or alternatively, *phenomenological models*. (see Multiphase Flow and Two-Phase, Gas-Liquid Flow)

Here, the phases are considered to be flowing separately in the channel, each with a given velocity and each occupying a given fraction of the channel cross section. The separated flow momentum equation reduces, for a duct of constant cross sectional area and steady flow, to:

where ε_{G} is the void fraction and ρ_{TP} is given by:

Again, the three terms on the right-hand side of Eq. (7) denote respectively the frictional, acceleration and gravitational pressure gradient terms. As will be seen from the above equations, calculation of the latter two terms (accelerational and gravitational) requires, in contrast to the homogeneous model, a value for the void fraction. Thus, to use the separated flow model, a void fraction correlation has to be invoked (see Void Fraction). The fractional pressure gradient (−dp_{F}/dz) (=τ_{0}P/A) is widely calculated in terms of the ratio of the two-phase frictional pressure gradient to the frictional pressure gradients for the liquid phase flowing alone in the channel, (dp_{F}/dz)_{L}, the gas flowing alone in the channel, (dp_{F}/dz)_{G}, or the frictional pressure gradient for the total flow flowing in the channel with liquid phase properties, (dp_{F}/dz)_{LO}. Pressure drop multipliers are then defined as follows:

The best known correlation in terms of such multipliers is that of *Lockhart and Martinelli* (1949) who ploted φ_{G} and φ_{L} as a function of the parameter X defined as follows:

Different curves were suggested, depending on whether the phase-alone flows were laminar (“viscous”) or turbulent, and the multipliers subscripted accordingly. The graphical correlation of Lockhart and Martinelli is shown in Figure 1.

Although still widely used, the Lockhart-Martinelli correlation has the disadvantage that it fails to predict adequately the effect of mass flux (and other parameters) and a whole variety of more sophisticated correlations have been produced as replacements (see Hewitt, 1982). Typical of these more recent correlations is that of Friedel (1979) whose correlation is given by:

where:

where σ is the surface tension and f_{GO} and f_{LO} are the friction factors for gas and liquid single phase flows at the total mass flow. This correlation has a standard deviation of around 30% for single Component flows and about 40-50% for two-component flows.

Prediction of pressure drop, is, as will have been seen, subject to large error. This error can be reduced if proper account is taken of the actual nature of the two-phase flows, namely the flow pattern or flow regime (see Two-Phase, Gas-Liquid Flow). Relationships are then developed for the various features of the flow and combined into an overall model which represents these features in on to predict the important design variables, such as pressure drop. Examples of such an approach are given in the articles on Annual Flow; Bubble Flow; Stratified Flow Gas Liquid and Slug Flow.

#### REFERENCES

Beggs, H. D. and Brill, J. P. (1973) A study of two-phase flow in inclines Pipes, *J. Petroleum Tch.*, 25, 607-617.

Friedel, L. (1979) Improved friction pressure drop correlations for horizontal and vertical two-phase pipe flow. European Two-Phase Flow Group Meeting, Ispra, Italy, paper E2.

Hewitt, G. F. (1982) Pressure drop. *Handbook of Multiphase Systems*, 2.2., G. Hetsroni,) ed., McGraw Hill Book Company, New York.

Hewitt, G. F. (1984) Two-phase flow through orifices, valves, bends and other singularities. Proceedings of the Eighth Lecture Series on Two-Phase Flow, University of Trondheim, 1984, 163-198.

Lockhart, R. W. and Martinelli, R. C. (1949) Proposed correlation of data for isothermal, two-phase, two-component flow in inpipes, *Chem. Prog*. 45, 39-48.

#### References

- Beggs, H. D. and Brill, J. P. (1973) A study of two-phase flow in inclines Pipes,
*J. Petroleum Tch.*, 25, 607-617. DOI: 10.2118/4007-PA - Friedel, L. (1979) Improved friction pressure drop correlations for horizontal and vertical two-phase pipe flow. European Two-Phase Flow Group Meeting, Ispra, Italy, paper E2.
- Hewitt, G. F. (1982) Pressure drop.
*Handbook of Multiphase Systems*, 2.2., G. Hetsroni,) ed., McGraw Hill Book Company, New York. - Hewitt, G. F. (1984) Two-phase flow through orifices, valves, bends and other singularities. Proceedings of the Eighth Lecture Series on Two-Phase Flow, University of Trondheim, 1984, 163-198.
- Lockhart, R. W. and Martinelli, R. C. (1949) Proposed correlation of data for isothermal, two-phase, two-component flow in inpipes,
*Chem. Prog*. 45, 39-48.