FLOW PATTERN TRANSITION IN TWO PHASE FLOW

by

Yehuda TaitelDepartment of Fluid Mechanics and Heat Transfer,

Faculty of Engineering,Tel-Aviv University, Ramat-Aviv 69978, ISRAEL

An extended Summary prepared for the2nd Annual meeting of the Institute of Multifluid Science and Technology,

March 18, 19, and 20, 1999

ABSTRACT

This paper presents the role of the flow patterns in two-phase gas liquid flow in pipes aswell as the methods used to predict the flow pattern transitions. Major milestones in thedevelopment of the main ideas concerning the prediction of the flow pattern are reviewed ina “historical” perspective.

The paper is limited to “steady state” flow, it is, the case where flow rate and inlet outletpressures are constant. Also transient phenomena like terrain slugging and void fractionwaves are not considered.

FLOW PATTERN CLASSIFICATION

The flow seen in two-phase pipe flow consists of numerous flow configurations that aremany times chaotic and difficult to classify. The main task is to group together the basicflow configurations and define a few basic patterns. This task is by no means well definedand indeed many configurations exist as individual researchers group the flowconfigurations somewhat differently depending on their own interpretation. Basically theattitude of most researchers is to minimize the number of flow pattern groups and to grouptogether the flow configurations that have basically the same character pertaining to thedistribution of the interfaces and the mechanisms dominating pressure drop and masstransfer.

The flow patterns can be grouped into four main classes where each class can besubdivided into sub-classes for more detailed description. The following classes areobserved:

(1) Stratified flow (Subclasses: stratified smooth, stratified wavy)(2) Intermittent flow (Subclasses: elongated bubble, slug, churn)(3) Annular flow (Subclass: wispy annular)(4) Bubble flow (Subclasses: bubbly, dispersed bubble)

THE DILEMMA OF FLOW PATTERN PREDICTION

Predication of the flow pattern is by no means an easy task. The main reason is due tothe complexity of the two-phase flow and the inability of rigorous methods to besuccessfully used here. In principle the two phase flow is governed by the hydrodynamicequations, that is, the Navier Stokes equations (for Newtonian fluids) and a solutionshould be "just" a mathematical procedure that solves the Navier Stokes equations for theliquid and the gas with the proper boundary conditions and the matching conditions at theinterfaces. However, this "straightforward" solution is, at least at the present time, and thenear future, clearly hopeless. The phenomena that take place in two-phase flow are muchtoo complex to be amenable to a rigorous mathematical solution. The main problem is theinability of the present methods to predict the distribution of the interfaces. This distributionis governed by a multitude of factors that include gravity, geometry, flow rates, fluidproperties and interfacial properties. In addition the two-phase flow is usually turbulentwhich makes a hopeless situation even worse.

In such a complex situation engineers avoid the mathematical difficulties by resorting toexperimental methods and develop "correlation" for engineering application. Thesecorrelations are based on experimental results but, when the number of parrameters thatcontrol the flow pattern is large, than even this basic problem has its difficulties.

Not less than 11 parameters can be identified to affect the flow pattern:

(1) The liquid superficial velocity, ULS [m/s] (it is customary to use the superficial velocity instead the flow rate).

(2) The gas superficial velocity, UGS [m/s].(3) Liquid density, ρL [Kg/m3].(4) Gas density, ρG [Kg/m3].(5) Liquid viscosity, µL [Kg/s m].(6) Gas viscosity, µG [Kg/s m].(7) Pipe diameter, D [m].(8) Acceleration of gravity, g [m/s2].(9) Surface tension, σ [Kg/s2].(10) Pipe roughness, ε [m].(11) Pipe inclination, β [0].

This shows that the flow pattern transition depends on 11 parameters or 8 dimensionlessvariables. Clearly, to find a relation among 8 parameters, based on experimental data aloneis quite impractical. This is the reason why the "straightforward" experimental technique isnot quite applicable for two phase-flow predictions. Nevertheless experimental data used tobe the dominant method for the prediction of flow pattern. It involved the collection ofexperimental data followed by mapping of the data in a two-dimensional plot by locatingtransition boundaries between the flow patterns. Such a plot is termed flow pattern map.This map intends to serve as the source by which prediction of the flow pattern for designpurposes take place.

METHODS USED FOR FLOW PATTERN PREDICTION

The first approach for the prediction of flow patterns is based on experimental data that areplotted on a ìflow pattern mapî. The earliest flow regime map is attributed to Baker (1954).Many more have since been suggested for horizontal, vertical and inclined pipes. A list ofcited references follows. For horizontal flow: White & Huntington (1955), Govier & Omer(1962), Al-Sheikh et al. (1970), Mandhane et al. (1974), Weisman et al. (1979) and

Spedding & Spence (1993). The vertical case is reported in: Griffith & Wallis (1961),Duns & Ros (1963), Sternling (1965), Wallis (1969), Hewitt & Roberts (1969), Govier &Aziz (1972), Wallis & Dobson (1973), Gould et al. (1974), Oshinowo & Charles (1974),Wisman (1975), Mishima & Ishii (1984), McQuilan & whalley (1985), Mukherjee & Broil(1985) and Bilicki & Kestin (1987). Vertical downwards flow was considered by:Crawford, et al. (1985, 1986). The inclination case is considered by: Weisman & Kang(1981), Spedding & Nguyen (1980) and Crawford & Weisman (1984).

A different approach was introduced by Taitel and Dukler (1976) for the case of horizontaland slightly inclined pipes and in (1980) for vertical flow. The vertical downwards flowand the inclined flow was considered by Barnea et al. (1982a,b, 1985).

The approach can be termed “mechanistic modeling”. In this procedure one should identifythe dominant physical phenomena that cause a specific transition. Then the physicalphenomena are formulated mathematically and transition lines are calculated and canpresented as an algebraic relation or with respect to dimensionless coordinates. Althoughthere is no guarantee that this method leads always to correct results the allegation is thatonce some experimental data verify the results based on this method then extrapolation todifferent conditions is much safer than those based solely on experimental correlation.

The shortcoming of the work of Taitel & Dukler (1976 and 1980) is that different modelswere used for horizontal and slightly inclined flows and for vertical flow. Some of thetransitions results from the same mechanism and should be based on the same physicalphenomenon (for example the transition to dispersed bubble and to annular flow). Barnea(1987) proposed a unified model in which one uses the same models for all inclinationangles and the effect of angle changes is automatically taken care by the proposed models.Table 1 is a summary of the transition mechanisms proposed by Barnea (1987).

Table 1: Summary of transition boundaries

Boundary Patterns Mechanism

A Stratified- nonstratified Kelvin- Helmholtz instabilityB From bubbly α>0.25C Stratified smooth-wavy Jeffreys, wind-wave interactionD From dispersed bubble Turbulent fluctuations and buoyancyG From dispersed bubble α >0.52H Slug- Churn α slug>0.52J From annular High void fraction or instabilityL Stratified wavy-annular Trajectory of torn dropsM Stratified smooth-wavy Fr>1.5N Elongated bubble-slug α slug =0• Minimum pipe diameter to Velocity of small bubbles versus

show bubbly flow elongated bubbles• Minimum inclination angle Lift versus buoyant forces to show bubbly flow

The strength of the mechanistic modeling lies also in its simplicity. All the above transitionsare easily calculated with minimal numerical efforts.

THE STRATIFIED-NON STRATIFIED TRANSITION

The transition from stratified to non-stratified attracted much more attention than the othertransition boundaries. The reason for this is perhaps owing to the importance of thistransition and\or to the mathematical amenability of this transition to be analyzed moreaccurately. Two basic approaches were used: (1) Analysis of the interfacial instability usingmore accurate (or sophisticated) approaches then the one proposed by Taitel and Dukler(1976) and Barnea (1987). (2) Analyzing the stability of slug flow based on the idea thatunder certain conditions the stratified-intermittent transition is caused not by interfacialinstability but due to inability of intermittent flow to sustain.

Interfacial Instability

Taitel & Dukler (1976) and Barnea (1987) and others used a simplified instability criterion,based in fact on the so called “Kelvin-Helmholtz” criterion for inviside flow. A morecomplete and accurate analysis is performed using rigorous stability analysis applied to thetwo-fluid model formulation. One can recognize four types of interfacial stability analyses:

(1) Viscous linear Kelvin-Helmholtz (VKH) analysis(2) Inviscide linear Kelvin-Helmhotz (IKH) analysis(3) Viscous non-linear Kelvin-Helmholz analysis(4) Structural stability analysis

In the viscous linear Kelvin-Helmhotz analysis (Wallis, 1969; Lin & Hanratty, 1986, 1987;Wu et al., 1987; Andritsos et al., 1987, 1989; Barnea, 1991, Crowley et al., 1992 andBarnea & Taitel, 1994a,b) a linear stability analysis is applied on the complete two fluidmodel. The results indicates when the interface is unstable. Instability for long wavelengthis attributed to the transition to intermittent flow provided the liquid holdup is large. That isthere is sufficient liquid in the stratified flow to support a stable slug. If not wavy structureis developed.

The Inviscide linear Kelvin-Helmholtz uses the same analysis but with the neglect of shearstresses, namely assuming that the fluid are inviscide. Barnea & Taitel (1993) claimed thatbeyond the instability boundary predicted by this transition stratified flow will never takeplace and the flow will be either intermittent or annular flow, depending on the liquidholdup.

The non linear stability analysis of the interface is carried out using numerical simulation bythe method of characteristics (Barnea & Taitel, 1994a,b) that examines the system responseto finite disturbances of the interface. The non linear simulation usually confirm theconclusions that are based on the linear considerations. It is shown that when the interfaceis linearly stable the simulation shows that waves decay with time. When the interface isunstable to the VKH the waves grow and reach a point where they become ill posed at thewave crest or reach a point of multi valued amplitude. The condition of ill posedness isassociated with an unbounded growth and transition to slug or annular flow (depending onhL/D). The condition of multivalued amplitude is interpreted as a wave break and it isbelieved to indicate a pattern of roll waves.

The structural stability analysis analyses the stability of the structure with respect to theaverage film thickness as obtained by the steady state solution, even if the interface isunstable (Barnea & Taitel, 1989, 1990). This analysis allows the prediction of theboundary from annular flow where the flow is separated although it is unstable with respectto Kelvin-Helmnoltz mechanism which results in a wavy interface. This analysis also helpsto distinguish between real physical and unstable steady state solutions that does not exist

(Barnea & Taitel 1992) when multi-valued steady state solutions are calculated.

Instability of slug flow

An alternative method for checking the stability of stratified flow is to analyze the stabilityof slug flow. When slugs are stable, slug flow will persist. If slugs are unstable they willdissipate and result in stratified flow. Three mechanisms are suggested to predict slugstability

(1) The elongated bubbles behind the slugs are slower than the front slug velocity(2) There is no solution to the slug flow hydrodynamic model.(3) The use of hydraulic jump criterion in the slug front

Bendiksen and Espedal (1992) proposed that a necessary condition, but not a sufficientcriterion, for the transition to “stable” slug flow is when the slug back (front of the bubblebehind the slug) is slower than its front, leading to the dissipation of slugs. The analysisassumes a stratified equilibrium film on which a spontaneous slug is formed. A massbalance in the slug front yields the front velocity while Bendiksen (1984) correlation isused for the bubble velocity. They noted that for low pressure the KH criterion is above(on a ULS vs. UGS map) the slug stability criterion. Thus slug flow will be observed abovethe KH boundary. But the region between the KH boundary and the slug stabilityboundary, slug flow can take place provided special entrance conditions are used to initiateslugs. On the other hand for high-pressure large diameter pipes the slug stability criterion isabove the KH criterion and thus the slug stability criterion is a more restrictive criterion thatcan be used alone for the prediction of the stratified-slug transition boundary.

Taitel et al. (1999) studied condition under which slug flow modeling has no solution(Taitel and Barnea, 1990). In this case slug flow will not exist. The physical interpretationhere is similar to the one given by Bendiksen (1992), that is, in this case the slug frontvelocity is slower than the slug back velocity. The results, at least for low pressureconditions is similar to the one given by Bendiksen (1992) (higher pressure was nottested), yet the analysis is quite different. For example Taitel et al consider only slughydrodynamics and did not consider stratified equilibrium flow in front of a slug.

A still different model for slug stability is introduced by Ruder et al (1989, 1990). Theyclaimed that the slug front is an hydraulic jump relative to the moving slug. the hydraulicjump criterion is based on a Froud number and the film in front of the slug is calculatedusing “Benjamin bubble” (Benjamin, 1968). As before, this is again a necessary conditionsfor the existence of slugs.

SUMMARY AND CONCLUSIONS

Method for the predication of flow pattern can be classified with respect to three categories.

(1) Experimental correlations.(2) Mechanistic modeling.(3) “Accurate analysis”.

The distinction between method (2) and (3) are not clear-cut. Both, in fact are one sort oranother of modeling and the “accurate” method is by no means a rigorous one. Thedifference probably lies in the degree of simplification and easy of calculation. Thus, forexample, Taitel & Dukler (1976) transition from stratified flow is a ìmechanisticî modelwhile the complete stability analysis of the two fluid model to calculate the transition basedon the VKH is an “accurate” approach.

It is somewhat frustrating that so far there is no accepted method that is considered thecorrect one to calculate the transition boundaries and that different mechanisms areproposed by different researchers for the same transition boundaries. Furthermore, evenexperimental results, reported by different researchers, differ in the location of thetransition boundaries also when conditions of the experiment are identical. Future workshould (1) further attempt to develop better and more accurate exact analyses and (2) tocontinue to look for the ìultimateî approximate solutions, based on ìmechanisticî modelingwhich are much more amenable to the practical engineer. In addition the task of looking formore data, especially those that will include higher pressure and large pipe sizes should becontinued since experiment is the only judge of the theoretical approaches.

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