When gas and liquid flow in a pipe, there are particular conditions for which the two phases are separated from each other by a continuous interface. Such a flow is dominated by the gravity force which causes the liquid to stratify at the bottom of the pipe (Figure 1). This flow pattern can be observed in horizontal or slightly inclined pipelines. It is characterized by the structure of the interface which may be smooth or wavy according to the gas flow rate. At low gas velocity, the interface is smooth or may be rippled by small capillary waves of a few millimetre length. With increasing gas velocity, small amplitude regular waves appear. At high enough velocity of gas, droplets can be entrained from the large amplitude, irregular waves and can be deposited at the wall or at the interface. However, this Atomization phenomenon is out of the scope of this entry (see Annular flow; Dispersed flow).
In stratified flow, the determination of pressure drop and liquid hold-up requires an accurate prediction of the friction at the wall and at the interface. Indeed, for fully developed flow, the pressure drop is controlled by the friction at the wall of each phase and by the weight of the liquid, which is related to the hold-up. As a consequence, pressure drop and liquid hold-up are strongly coupled in stratified flow: they must be predicted simultaneously. Another interesting feature of stratified flow is that the difference of velocity between phases can be high, suggesting that the momentum transfer between phases is ineffective as a mechanism for the gas to drive the liquid. As this transfer is controlled by the interfacial friction, it may be anticipated that it will play a central role in the flow modelling.
At low enough velocity the interface is maintained smooth by gravity and surface tension. When the gas velocity increases waves develop at the surface through a Kelvin-Helmholtz Instability mechanism. These waves act upon the gas as interfacial roughness and the interfacial stress becomes larger than over a smooth and flat solid surface. That interfacial friction depends on interfacial roughness, and roughness itself depends on the phase velocities, makes the problem difficult to solve. This is a twofold problem which is regarded as the central issue of stratified flow modelling.
Different approaches have been explored to solve this problem. An empirical, but effective, approach correlates the interfacial stress with the mean phase velocities and fluid properties: by similarity to single-phase turbulent flow over rough surfaces, it is possible to correlate the interfacial friction factor to the wave roughness experienced by the gas. A more recent approach involves numerical flow simulation to predict the wave drag over simplified interfacial shapes, the monochromatic wave being the simplest case to be studied. As pointed out by Hanratty and McCready (1992), “a critical physical problem is to reconcile these approaches so as to produce a unified theory” of interfacial transfer of momentum in stratified two-phase flows.
In the first part of the entry, wave generation and wave pattern are presented. In the second part, modelling of wavy stratified flow is introduced. The closure relations of wall and interfacial friction coefficients are given in the third part. In the last part, an alternative approach of interfacial momentum transfer modelling is discussed.
Cohen and Hanratty (1965) have shown that the waves receive their energy from the gas through the work of pressure and shear stress perturbations induced by the waves. Andritsos (1986) observed that for a low-viscosity liquid, the first waves generated are small-amplitude regular two-dimensional waves: they receive energy from the pressure perturbations in the gas which are in phase with the wave slope. The energy transfer being induced by a sheltering mechanism, these waves will be called J-waves, referring to Jeffrey’s sheltering hypothesis (1925). With increasing gas velocity and for given liquid flow rate, the wave amplitude becomes larger and the wavelength decreases: hence the wave steepness increases. These waves receive energy from the pressure perturbations in the gas which are in phase with the wave height. Surprisingly, Andritsos found that these large-amplitude waves are generated at flow conditions corresponding to the occurrence of Kelvin-Helmholtz instability. These waves will be thus called KH-waves. At higher gas velocity, irregular large-amplitude waves are formed with steep front and gradually sloping back: they are called roll waves. They may break for large enough amplitude. Roll waves are widely spaced from each other with intermittent occurrence. There is no theoretical prediction of this intermittence. Only a few measurements are available (Vlachos et al., 1993 and Strand, 1993). The wave pattern depends on the liquid viscosity: Andritsos has observed that the domain of J-waves vanishes for high viscosity liquids.
The initiation of J-waves can be estimated for low viscosity liquids by the criterion proposed by Taitel and Dukler (1976):
where u, ρ, ν are velocity, density, kinematic viscosity, g is acceleration due to gravity and s is a sheltering coefficient which is about 0.06. As indicated by Andritsos the transition to KH-waves coincides with the inviscid Kelvin-Helmholtz relation:
where k is the wave number, σ the surface tension, h the liquid height and j the volumetric flux. The critical value corresponds to the wave number k for which the velocity is minimum:
The wave patterns are plotted for a large range of gas and liquid superficial velocities in Figure 2.
Figure 2. Wave patterns in stratified flow in a pipe after Lopez (1994). smooth interface, 2D waves, 3D waves, roll waves.
The magnitude of the wave drag increases with the amplitude and the wave number. For KH-waves, a large increase of the interfacial transfer is observed (Andritsos, 1986, Strand, 1993, Lopez, 1994) suggesting that the drag must be related to the wave steepness. (See also Waves in Fluids and Wavy Flow.)
Suppose that we know the specific flow conditions, namely:
the pipe size, i.e., its diameter D or its cross-section area A = πD^{2}/4, and the inclination θ,
the fluid properties, i.e., the densities ρ_{k} (k = L, G), the kinematic viscosities ν_{k} and the surface tension σ,
the superficial velocity— or volumetric flux —of each phase, j_{k} = Q_{k}/A.
A useful model of stratified flow would produce at least the following informations:
the mean velocities of both gas and liquid μ_{k},
the phase fractions of both gas and liquid ε_{k},
the wall and interracial stresses,
the pressure gradient dp/dx.
For sake of simplicity, let us restrict the study to fully-developed steady flow and assume that the gas is incompressible. The mass balance for each phase reduces to:
where is the volumetric flow rate. The phase fractions are related by the trivial expression:
The momentum balance in each phase and across the interface may be written:
where P_{w} and P_{i} are the perimeters of the wall and the interface, τ_{w} and τ_{i}, the Shear Stresses at the wall and the interface and θ, the pipe inclination. The equations for the pressure gradient dp/dx and the liquid hold-up ε_{L} may be obtained from the two momentum equations:
where P = P_{wL} + P_{wG} is the total perimeter wetted by the phases, τ_{w} = (P_{wL}τ_{wL} + P_{wG}τ_{w})/P is the total shear stress and ρ_{m} = ρ_{G} ε_{G} + ρ_{L} ε_{L} the density of the mixture.
The wall and interfacial perimeter can easily be related to the phase fractions. For pipe of circular cross section, they are easily expressed versus the angle Ω which intercept the interface:
The basic question is the modelling of wall and interfacial shear stresses usually expressed through dimensionless friction factors defined as:
The line followed by most authors to deduce the correlations for the wall and interfacial friction factors requires some attention. In stratified flow, only pressure drop and liquid hold-up can be measured easily. It is thus impossible to deduce from Eqs. 8 and 9 the experimental values of the three shear stresses τ_{wG}, τ_{wL}, τ_{iG}. It is generally assumed that the wall shear stress exerted by the gas is correctly predicted by a single-phase flow relation. Once τ_{wG} has been estimated, it is possible to deduce the experimental values of τ_{wL}, τ_{iG}. Fabre et al. (1987) were probably the first to measure the wall shear stress in the gas. Following their conclusion, the use of a single-phase flow relationship is acceptable to predict the wall friction factor for the gas phase. This relation depends on whether the flow is laminar or turbulent. It is extended to two-phase flow as follows:
in which k_{w} is the sand roughness of the wall, Dh and Re the hydraulic diameter and the Reynolds number defined for the gas as:
Similar method may be used for the shear stress of the liquid phase at the wall with a little difference: the hydraulic diameter does not include the interfacial perimeter
The extension of single-phase flow relation in the liquid phase leads however to some bias (Fabre et al., 1987; Rosant, 1984; Andritsos, 1986). Cheremisinoff and Davis (1979), Rosant (1984) and Andritsos have proposed empirical correlations for fwL. (See also Pressure Drop, Single Phase and Two-Phase.)
Nevertheless, it must be kept in mind that the crucial problem remains the closure of the interfacial friction factor f_{i}. Taitel and Dukler (1976) presented a two-fluid model based on balance equations, simplifying the problem by considering that f_{i} = f_{wG}. In this case, the gas-liquid interface is implicitly smooth and the interfacial stress is underestimated, leading to an overestimation of the liquid hold-up ε_{L}. This is in contradiction with the fact that the interfacial friction factor increases with the gas Reynolds number (Figure 3).
Figure 3. Interfacial friction factor vs gas Reynolds number after Lopez. j_{L} = 0.0066 m/s, j_{L} = 0.02 m/s, j_{L} = 0.05 m/s, ___ Eq. (14).
Among all the empirical correlations which have been developed for the interfacial friction factor, this of Andritsos and Hanratty (1987) must be recommended as it gives the best results. These authors postulated that there exists a critical gas flux j_{G,KH} given by Eq. (3) below which the interface is hydraulically smooth. Above this critical flux the interface is wavy and the interfacial shear stress is assumed to increase linearly with the difference j_{G} - j_{G,KH}
where f_{i,smooth} is the friction factor for a smooth interface calculated from Eq. (14) with k_{w} = 0. The interfacial friction factor increases when the gas velocity is high enough to generate KH-waves. It must be pointed out that even if it is disturbed by J-waves, the interface is considered to behave like a smooth surface.
In the previous approach, the wavy structure is only considered through the transition between the smooth and wavy regimes. Another approach considers that the interface is seen by the gas as a surface whose roughness changes with both gas and liquid velocities. If we accept that the momentum transfer across a rough liquid surface is governed by the same mechanism as for a rough wall, it is possible to extend the single-phase flow correlation given by Eq. (14):
The problem is completely solved provided that the roughness of the interface k_{i} may be predicted.
A solution ignoring the wave amplitude has been proposed early on by Charnock (1955) for fully developed wave field in deep water. For inviscid fluids, only gravity and pressure forces balance yielding:
in which u_{*i} = (|τ_{iG}|/ρ_{G})^{1/2} is the interfacial friction velocity and γ a constant within the range 0.1–0.5. Equation (21) has to be solved together with Eq. (20). By taking into account the definition (Eq. (12)), one obtains the following implicit relation which must be solved with an iterative procedure:
where Fr_{G} is a Froude Number defined as:
Another method involves the wave structure. Indeed the interfacial roughness must depend on different length scales of the wave field. For periodic waves these length scales are the amplitude and the wavelength. For random waves in rectangular channel, Cohen and Hanratty (1968) found that the roughness depends on the r.m.s. of the instantaneous liquid height:
However, only waves which emerge from the viscous sublayer can create roughness, so that the correlation of Cohen and Hanratty (1968) must be corrected as suggested by Fabre et al. (1987). Nevertheless, this approach does not provide the key to a successful prediction of the interfacial roughness since the prediction of the wave length scales remains an open problem.
In particular, is related to the distribution of energy among all the wave frequencies ω:
Bruno and McCready (1989) and Jurman and McCready (1989) attempted to determine the wave energy spectrum. Their analysis is based on both energy transfer between wind and waves and nonlinear energy transfer between wave components.
Considering the complexity of the wave spectrum, one can simplify the wave field, and focus on the drag of a monochromatic wave. The interfacial transfer accounts for the viscous drag and the form drag. The viscous drag can be expressed through a classical friction factor correlation. The form drag can be expressed through a Drag Coefficient C_{D} accounting for the distribution of pressure over the wave surface. It can be determined analytically from a linear perturbation method, for a wave of small steepness ak, a being the amplitude and k the wave number (Liné & Lopez, 1995):
In this equation, |p| and θ_{p} are the modules and the phase angle of the pressure perturbation at the interface. The modules and the phase angle of the pressure can be estimated from the solution of perturbed flow model above a sinusoidal wave (Abrams, 1984; Harris, 1993; Lopez, 1994) or by direct numerical simulation (Hudson, 1993). However, following Jeffreys (1925), CD is proportional to the square of the steepness ak:
where s is a sheltering coefficient. In a recent theoretical work, Belcher and Hunt (1993) confirmed the relation of the drag coefficient above slowly moving waves as a function of the square of the steepness. In this case, the problem remains to model the characteristics of the dominant wave.
The fundamental issue remains to unify these different approaches of interfacial transfer so as to produce a reliable modelling of the interfacial transfer in separated two-phase flows.
Conclusion
Since the earliest model of Taitel and Dukler (1976), the improvement of gas-liquid stratified flow modelling has been related to the pertinence of the closure relations for the momentum transfers at the interface and at the wall.
On the one hand, the phase fractions and velocities depend on the interfacial momentum transfer, depending itself on the deformations of the gas-liquid interface. On the other hand, the deformations of the interface result from the interactions between gas and liquid. Indeed, the wave field propagates over the surface of the liquid; hence, it cannot solely result from local interactions between gas and liquid flows. Considering this complex problem, two approaches have been developed, the first being based on empirical correlations and the second on a local analysis of the flow. The first approach is useful at solving practical problems, the second is necessary to understand the physical mechanism governing the interfacial interactions.
The two-fluid model presented in the second part of this entry coupled to the closure relations discussed in the third part has the capability of predicting pressure gradient and liquid hold-up in stratified flow.
The information given in the fourth part can be summarized as follows:
the wave field over the liquid can be considered as an equivalent roughness, the problem being to predict this roughness: none of today’s solutions appears satisfactory;
advanced work concerning the modelling of the wave spectrum is taking place; however, this complex problem is not yet solved;
in the case of a dominant wave, the contributions of the form drag separate from the viscous drag can be estimated; however, the form drag depends on the characteristics of the dominant wave.
Consequently, an improvement of stratified flow models requires a better prediction of wave characteristics and a better understanding of the interaction between “wind” and waves.
REFERENCES
Abrams, J. (1984) Modeling of Turbulent Flow over a Wavy Surface, Ph. D. Thesis, Univ. of Illinois, Urbana, U. S. A.
Andritsos, N. (1986) Effect of Pipe Diameter and Liquid Viscosity on Horizontal Stratified Flow, Ph. D. Thesis, Univ. of Illinois, Urbana, U. S. A.
Andritsos, N. and Hanratty, T. J. (1987) Influence of interfacial waves in stratified gas-liquid flows, AIChE J., 33: 444-454.
Belcher, S. E. and Hunt. J. C. R. (1993) Turbulent shear flow over slowly moving waves, J. Fluid Mech., 251: 109-148.
Bruno, K. and McCready, M. J. (1989) Processes which control the interfacial wave-spectrum in separate gas-liquid flows, Int. J. Multiphase Flow, 15: 531-552. DOI: 10.1016/0301-9322(89)90052-9
Charnock, H. (1955) Wind stress on a water surface, Q. J. Met. Soc., 81: 639-640.
Cheremisinoff, N. and Davis, E. J. (1979) Stratified turbulent-turbulent gas-liquid flow, AIChE J., 25: 48-56.
Cohen, L. S. and Hanratty, T. J. (1965) Generation of waves in the concurrent flow of air and liquid, AIChE J., 11: 138-144.
Cohen, L. S. and Hanratty, T. J. (1968) Effect of waves at gas-liquid interface on a turbulent air flow, J. Fluid Mech., 31: 467.
Fabre, J., Masbernat, L., Fernandez-Flores, R., Suzanne, C. (1987) Stratified flow, Part II: interfacial and wall shear stress. Multiphase Science and Technology, Volume 3, G. F. Hewitt, J. M. Delhaye and N. Zuber, eds. Hemisphere.
Fernandez-Flores, R. (1984) Etude des interactions dynamiques en écoulement diphasique stratifié, Thèse de Docteur Ingénieur, I. N. P. Toulouse, France.
Hanratty, T. J. (1983) Interfacial instabilities caused by air flow over a thin liquid layer, Waves on Fluid Interface, Academic Press, Inc., New York.
Hanratty, T. J. and McCready, M. J. (1992) Phenomenological Understanding of Gas-Liquid Separated Flows, Proceedings of the Third International Workshop on Two-Phase Flow Fundamentals, Imperial College, London, U. K., April 1992.
Harris, J. A. (1993) On the Growth of Water Waves and the Motion Beneath Them, Ph. D. Thesis, Stanford Univ., U.S.A.
Hudson, J. D. (1993) The Effect of a Wavy Boundary on a Turbulent Flow, Ph. D. Thesis, Univ. of Illinois, Urbana, U.S.A.
Jurman, L. A. and McCready, M. J. (1989) Study of waves on thin liquid films sheared by turbulent gas flow, Phys. Fluids, A1–3: 522-535.
Liné, A. and Lopez, D. (1995) Two-fluid Model of Separated Two-Phase Flow: Momentum Transfer on Wavy Boundary, Proceedings of Second International Conference on Multiphase Flow, Kyoto, Japan.
Lopez, D. (1994) Ecoulements diphasiques a phases séparées à faible contenu de liquide, Thèse de Doctoral, I.N.P. Toulouse, France.
Rosant, J. M. (1984) Ecoulements diphasiques liquide-gaz. en conduile circulaire, Thèse de Doctorat ès-Sciences, ENSM, Nantes, France.
Sinai, Y. L. (1985) Interfacial phenomena of fully-developed, stratified, two-phase flows, Encyclopedia of Fluid Mechanics, Vol. 3, Gas-Liquid Flows, 475–491, N. P. Cheremisinoff, Ed.
Strand, O. (1983) An Experimental Investigation of Stratified Two-Phase Flow in Horizontal Pipes, Ph. D. Thesis, Univ. of Oslo, Norway.
Taitel, Y. and Dukler, A. E. (1976) A theoretical approach to the Lockhart-Martinelli correlation for stratified flow, Int. J. Multiphase Flow, 2: 591-595.
Taitel, Y. and Dukler, A. E. (1976) Model for predicting flow regime transitions in horizontal and near horizontal gas-liquid flow, AIChE J., 22: 47-55.
Vlachos, N., Paras, S. V., Karabelas, A. J. (1993) Liquid Layer and Wall Shear Stress Characteristics in Stratified-Atomization Flow, Proc. European Two-Phase Flow Group Meeting, Hanover, RFA.
Referências
- Abrams, J. (1984) Modeling of Turbulent Flow over a Wavy Surface, Ph. D. Thesis, Univ. of Illinois, Urbana, U. S. A.
- Andritsos, N. (1986) Effect of Pipe Diameter and Liquid Viscosity on Horizontal Stratified Flow, Ph. D. Thesis, Univ. of Illinois, Urbana, U. S. A.
- Andritsos, N. and Hanratty, T. J. (1987) Influence of interfacial waves in stratified gas-liquid flows, AIChE J., 33: 444-454.
- Belcher, S. E. and Hunt. J. C. R. (1993) Turbulent shear flow over slowly moving waves, J. Fluid Mech., 251: 109-148. DOI: 10.1017/S0022112093003350
- Bruno, K. and McCready, M. J. (1989) Processes which control the interfacial wave-spectrum in separate gas-liquid flows, Int. J. Multiphase Flow, 15: 531-552. DOI: 10.1016/0301-9322(89)90052-9
- Charnock, H. (1955) Wind stress on a water surface, Q. J. Met. Soc., 81: 639-640. DOI: 10.1002/qj.49708135027
- Cheremisinoff, N. and Davis, E. J. (1979) Stratified turbulent-turbulent gas-liquid flow, AIChE J., 25: 48-56.
- Cohen, L. S. and Hanratty, T. J. (1965) Generation of waves in the concurrent flow of air and liquid, AIChE J., 11: 138-144. DOI: 10.1002/aic.690110129
- Cohen, L. S. and Hanratty, T. J. (1968) Effect of waves at gas-liquid interface on a turbulent air flow, J. Fluid Mech., 31: 467. DOI: 10.1017/S0022112068000285
- Fabre, J., Masbernat, L., Fernandez-Flores, R., Suzanne, C. (1987) Stratified flow, Part II: interfacial and wall shear stress. Multiphase Science and Technology, Volume 3, G. F. Hewitt, J. M. Delhaye and N. Zuber, eds. Hemisphere.
- Fernandez-Flores, R. (1984) Etude des interactions dynamiques en Ã©coulement diphasique stratifiÃ©, ThÃ¨se de Docteur IngÃ©nieur, I. N. P. Toulouse, France.
- Hanratty, T. J. (1983) Interfacial instabilities caused by air flow over a thin liquid layer, Waves on Fluid Interface, Academic Press, Inc., New York.
- Hanratty, T. J. and McCready, M. J. (1992) Phenomenological Understanding of Gas-Liquid Separated Flows, Proceedings of the Third International Workshop on Two-Phase Flow Fundamentals, Imperial College, London, U. K., April 1992.
- Harris, J. A. (1993) On the Growth of Water Waves and the Motion Beneath Them, Ph. D. Thesis, Stanford Univ., U.S.A.
- Hudson, J. D. (1993) The Effect of a Wavy Boundary on a Turbulent Flow, Ph. D. Thesis, Univ. of Illinois, Urbana, U.S.A.
- Jurman, L. A. and McCready, M. J. (1989) Study of waves on thin liquid films sheared by turbulent gas flow, Phys. Fluids, A1â€“3: 522-535. DOI: 10.1063/1.857553
- LinÃ©, A. and Lopez, D. (1995) Two-fluid Model of Separated Two-Phase Flow: Momentum Transfer on Wavy Boundary, Proceedings of Second International Conference on Multiphase Flow, Kyoto, Japan.
- Lopez, D. (1994) Ecoulements diphasiques a phases sÃ©parÃ©es Ã faible contenu de liquide, ThÃ¨se de Doctoral, I.N.P. Toulouse, France.
- Rosant, J. M. (1984) Ecoulements diphasiques liquide-gaz. en conduile circulaire, ThÃ¨se de Doctorat Ã¨s-Sciences, ENSM, Nantes, France.
- Sinai, Y. L. (1985) Interfacial phenomena of fully-developed, stratified, two-phase flows, Encyclopedia of Fluid Mechanics, Vol. 3, Gas-Liquid Flows, 475â€“491, N. P. Cheremisinoff, Ed.
- Strand, O. (1983) An Experimental Investigation of Stratified Two-Phase Flow in Horizontal Pipes, Ph. D. Thesis, Univ. of Oslo, Norway.
- Taitel, Y. and Dukler, A. E. (1976) A theoretical approach to the Lockhart-Martinelli correlation for stratified flow, Int. J. Multiphase Flow, 2: 591-595.
- Taitel, Y. and Dukler, A. E. (1976) Model for predicting flow regime transitions in horizontal and near horizontal gas-liquid flow, AIChE J., 22: 47-55. DOI: 10.1002/aic.690220105
- Vlachos, N., Paras, S. V., Karabelas, A. J. (1993) Liquid Layer and Wall Shear Stress Characteristics in Stratified-Atomization Flow, Proc. European Two-Phase Flow Group Meeting, Hanover, RFA.