Some Issues Related to the Modeling of the Volumetric Interfacial Area in Gas-Liquid Two-Phase Flows

Abstract

During the last decades, the so-called two-fluid model has been the most widely used in two-phase flow studies for environmental and industrial applications. In this model, one set of mass, momentum and energy balance equations is written for each phase, therefore the model is able to deal with mechanical and thermal imbalances. The two phases cannot evolve independently, since they are coupled together through interfacial interaction terms representing the average exchanges between the two phases. The success of the two-fluid model in particular situations strongly depends on the modeling of these interfacial interaction terms. The interfacial interaction terms generally involve the volumetric interfacial area, that is the contact area between phases per unit volume of the two-phase mixture. This interfacial area is very important to express correctly the interfacial exchanges of mass, momentum and energy, and also gives an additional information about the interfaces structure, hence on the flow regime (bubbly, droplet, flow with separated phases…). Therefore, considerable attention has been focused on this subject in the last ten years. Theoretical and experimental researches have been done, especially on the bubbly flow configuration. Despite all these efforts, some issues remain and a general model giving the volumetric interfacial area in all two-phase flow regimes is not yet available. In the subject of the interfacial area modeling, three types of issues remain : theoretical, experimental and numerical ones. In what follows, we examine these three points independently in the general context of all two-phase flow regimes. From the theoretical point of view, most of the authors working on the subject try to write a seventh balance equation for the volumetric interfacial area, in addition to the six balance equations of the two-fluid model. This method seems promising, since it is able to deal with complex situations where different phenomena responsible for the flow regimes transitions, like coalescence, break-up, phase change and so on, act together. The transport of the VIA by the flow is also taken into account in this equation. However, such an equation is very difficult to establish in the general case where the flow regime is not known a priori. One should distinguish the relatively simple case of dispersed two-phase flows (bubbly and droplet flows) from the other, more complex, cases like the stratified flow, the annular flow or the churn-turbulent flow. For the dispersed flow case, an analogy with the kinetic theory of gases is generally adopted. The bubbles or droplets are described by a probability density function, as for the molecules in the kinetic theory of gases. A Liouville type equation is written for the pdf and the VIA balance equation is deduced from this equation. This method is very advantageous for the dispersed flow case since the coalescence and break-up terms can be introduced in a natural way. These terms correspond to the collisions terms in the kinetic theory of gases. The transport velocity appearing in the VIA balance equation is also clearly defined in this approach : it corresponds to the centre of area velocity of the particles swarm. Unfortunately, this method cannot be easily extended to the other flow regimes, especially the flows with separated phases like the annular or the stratified flows. To make the analogy with the kinetic theory of gases, one needs to introduce a population balance, and there is no population in the stratified or the annular flows, since only one continuous interface exists. Several attempts to establish a general balance equation not restricted to a particular flow regime have been done. But it appeared to the authors that these balance equations are nothing else than a particular form of the Leibniz rule for the surfaces. Therefore the application of these equations to the modeling of the VIA in two-phase flows is highly questionable. The issue of the establishment of a general balance equation for the VIA, able to deal with all two-phase flow regimes, remains open. From the experimental point of view, new instrumentation is available today to measure the volumetric interfacial area. The most promising method seems to be the use of local resistive or optical probes. Four-sensor probes are theoretically able to measure the local volumetric interfacial area whatever the flow regime if the following requirements are satisfied. The interfaces must always move in the flow and the smallest radius of curvature of the interfaces passing through the probe must be significantly larger than the probe size. The method has the inconvenient to be intrusive, but non intrusive methods, like the photographic method, are generally restricted to particular flow regimes like a bubbly flow regime characterised by low values of the void fraction. The two-sensor and four-sensor probes have been tested with success in the bubbly flow regime by several authors, up to the cap bubbly flow. We will try to use a four-sensor probe in the case of a stratified wavy flow in the CEA Grenoble. The application of these measuring techniques to the droplet flow case seems more difficult because of the small size of the droplets generally encountered, the high velocity of the droplets in a gas stream and the possibility that the droplets form liquid films on the probe. Therefore, for droplet flows, the use of a photographic method is perhaps preferable. From the numerical point of view, the use of a volumetric interfacial area balance equation in a code and the determination of the local flow regime from the calculated VIA necessitates a lot of modeling efforts, and numerous iterative comparisons to the experiments. In our opinion, the major difficulties are the prediction of a stratification or a destratification (transition from a dispersed flow regime to a flow regime with separated phases and vice-versa), and the phase inversion (when a bubbly flow becomes a droplet flow and vice versa). The VIA in a bubbly flow is typically much greater than the one in a stratified flow, and the source and sink terms governing the transition between these two regimes (for example the deposition rate of the bubbles on the free surface) are not easy to model. During a phase inversion (governed for example by a criterion based on the local value of the void fraction), the source terms of VIA for bubbles and droplets are not the same, and this can bring some discontinuities on the VIA during the calculation. This problem is due to the fact that the same balance equation is used for all the flow regimes. A solution to this problem could be to introduce several interfacial area balance equations : one for the bubbles surfaces, a second one for the droplets surfaces and possibly a third one for the free surface. This method has been used with success in the SIMMER code, where separated VIA balance equations are used for bubbles and droplets, but we believe that it is more difficult to use these VIA to determine the flow regime. At the moment, numerous authors use a VIA balance equation in their computer code for the case of bubbly-to-slug flow regimes. The issue of the possible extension of these balance equation to the other flow regimes should be addressed in the future.