Stability Parameter for Two-Phase Vertical Annular Flow
Authors: Bordalo S.N. , Gaspari E.F. 
Published: December 2007

Abstract
A good theory of stability is essential to the prediction of flow pattern transition; therefore, it is crucial to the design of two-phase flow systems, since the behavior and performance of such systems (flow rate versus driving head) is inherently dependent on the actual pattern that occurs in the flow (bubbles, slugs, annular, stratified etc.). The objective of this work is to analyze the stability of two-phase vertical annular flow, and to identify a parameter that governs the transition to slug or churn flow. Previous theoretical investigations, found in the literature, were inconclusive or showed results inconsistent with experimental data. Briefly, for those studies, the paradox of vertical annular flow was: "the flow is unstable, therefore it should not exist… but it does". In the present work the analysis is conducted through two methods - the first method uses a transient simulator, that models the two phases separately; the second method uses linear analysis akin to Kelvin-Helmholtz analysis. For both methods, the effect of the gas compressibility is included. The gas compressibility was neglected by most of the previous studies. The results demonstrate that the gas compressibility is the main governing parameter of stability. Calculations of stable conditions showed very good agreement with the experimental maps of two-phase flow patterns, for pipe inclinations of 50, 150, 300, and 900 (vertical).

Introduction
The objective of this work is to determine the conditions under which two-phase annular vertical flow is stable, and to establish which parameters govern the stability. The results of this study offer a significant contribution to the theory of the dynamics of two-phase flow. and are of great importance to the practice of petroleum and nuclear engineering
Maps of Flow Patterns. Fig. 1 and Fig. 2 present examples of empirical maps of flow patterns for liquid-gas flow. The production of such maps requires an enormous investment of material and human resources, with costs that are an obstacle to a complete and proper study of the behavior and effect of all the significant flow variables, for the broad working range of their values. A good theory is needed to avoid such costs. Fig. 3 and Fig. 4 illustrate some of the transitions undergone by the annular flow when the liquid film cannot be sustained - transition to the slug pattern, in the horizontal case, and transition to the slug or to the churn pattern, in the vertical case.
Previous Studies. One of the first works of analysis of interfacial stability was undertaken by Jefreys (1925) for channel flow. Wallis (1973) studied the stability of the stratified pattern. Taitel and Dukler (1976) used an inviscid model for both phases of a horizontal flow. Hanratty (1983), Lin and Hanratty (1986), Wu et al. (1987), Crowley et al. (1992), included viscous effects in their analysis of horizontal flows. Barnea and Taitel, (1993, 1994) combined the inviscid and viscous linear stability criteria of Kelvin-Helmholtz (IKH and VKH) to analyze the stratified horizontal flow. In the case of vertical pipes, the work of Barnea and Taitel (1994) led to the conclusion that the liquid-gas interface of annular flow is unconditionally unstable, for reasons that will be discussed below; as a consequence, a theory of structural stability was devised to explain the "paradoxical" stable existence of annular vertical flow in many engineering systems. All the aforementioned studies used models based on equations for one-dimensional flow.
Physics of the Interface. When a disturbance deforms the liquid film interface, increasing the film thickness as shown in Fig. 5, the gas streamlines tend to converge, the gas velocity tends to increase and the pressure in the gas phase tends to drop (due to the interchange between kinetic and pressure energies - the "Bernoulli effect"); meanwhile, the inverse happens in the liquid phase. P. 663^