Two-phase Flow Solution Methods

By default, Ansys CFX is able to solve a two-phase flow problem by using the Eulerian- Eulerian model or the Lagrangian Particle Tracking model. Of interest for this work is the Eulerian-Eulerian approach. Within the Eulerian-Eulerian formulation certain interphase transfer terms used in the momentum, heat and other interphase transfer models can be modeled using the [Ansys 2009]:

 Particle Model.

 Mixture Model.

 Free Surface Model.

Also the interfacial area density, for the inhomogeneous transfer between a pair of fluids, is calculated according to one of the three models above listed [Ansys 2009]. The simulation is based on a two-fluid approach, already introduced in chapter 4.1. If the Particle Model is used, then the two phases are considered to be one continuous and other dispersed. The dispersed phase can be considered as:

 Mono-dispersed

 Poly-dispersed (MUSIG Multiple Size Group)

If the gas is considered to be the dispersed phase, in the monodispersed approach, all the bubbles are supposed to have the same spherical form and the same average

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diameter. Their common diameter can be a constant or an expression dependent on local parameters. Furthermore, they share the same dispersed phase velocity field. In this case for a two-fluid system, the equations to be solved are in total 11 divided as follows:

 Momentum of the liquid phase: 3  Momentum of the gas phase: 3  Pressure: 1

 Mass of liquid: 1  Mass of gas: 1

 Turbulence in the liquid phase: 2

In the polydispersed approach, if the gas is supposed to be the dispersed phase, all the bubbles present in the system are supposed to share the same spherical form, but not the same average diameter. In this case, the bubbles are divided in classes and the initial bubble size distribution is defined as a boundary condition. The bubble groups can share the same velocity field or they can be associated to different velocity field. The first case refers to the homogeneous model where as the second case refers to the inhomogeneous model. The intergroup bubble transfer is simulated as explained previously in chapter 4 by means of coalescence and breakup source and sink terms. They determine the changes in the bubble group size fractions as a result of break-up and coalescence process. In the case of a two-fluid system (homogeneous MUSIG) considering n bubble size groups the equations to be solved are in total 11+n divided as follows:

 Momentum of the liquid phase: 3  Momentum of the gas phase: 3  Pressure: 1

 Mass of liquid: 1  Mass of gas: 1

 Turbulence in the liquid phase: 2  Bubble group size fraction: n

If also cap bubbles or slugs exist, then a third fluid (gas) needs to be introduced with adequate transfer terms between with the other two. In fact, when the flow conditions are changing and bubbles of different diameters and shapes are formed, the definition of a second gas flow is justified because the interfacial structures in different flow regimes change dramatically [Ishii and Hibiki 2006]. Furthermore, the bubble

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interaction mechanisms in such flow conditions, driven by two bubble types, can be quite different compared with those considered by the other gas flow.

In the case of such a three-fluid system (inhomogeneous MUSIG) considering n bubble size groups (they can be even more than the previous case) the equations to be solved are in total 15+n divided as follows:

 Momentum of the liquid phase: 3  Momentum of the first gas phase: 3  Momentum of the second gas phase: 3  Pressure: 1

 Mass of the liquid phase: 1  Mass of the first gas phase: 1  Mass of the second gas phase: 1  Turbulence of the liquid phase: 2  Bubble group size fractions: n

The limit of the monodispersed approximation is that the average bubble size needs to

be known “a priori” and the shape of the bubbles needs to be nearly spherical. Furthermore, this approach is suitable for weak interacting flows with a particle size distribution concentrated around the average value. Of course, the bubble size can be defined as an algebraic expression related to local variables as, for example, the elevation. In this way it is possible to take into account the bubble expansion due to the decrease of hydrostatic pressure (the bubble rise in a bubble column or in a vertical upward pipe flow).

In case of the polydispersed approximation, it is possible to obtain very precise results depending on the quality of the interaction mechanism models used for the simulation. In practice, more than 15 bubble groups are needed to describe the flow accurately. This aspect is relevant because, if, on one side, the accuracy of the results increases compared to monodispersed simulation, on the other side, also the computational effort raises dramatically.

A good compromise between the monodispersed and polydispersed (homogeneous or inhomogeneous) methods is represented by the Interfacial Area Transport Equation approach. This allows to keep the computational effort as low as possible (essentially equal to the monodispersed case if only one additional transport equation is added), while taking into consideration the local bubble interaction mechanisms. Furthermore,

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if assumptions concerning the form of the bubbles present in the system are made, it is possible to calculate a local value for the bubble mean diameter. This is done by using simple algebraic relations between the volume and the interfacial area in each computational node.

This last simulation strategy is not, by default, included in Ansys CFX. However, it is possible for the user to define an additional transport equation associated to a user defined interfacial area variable. Furthermore, it is possible to define source and sink terms to modify the value of the transported quantity along the path.

One or two additional transport equations can be added to the original ANSYS CFX solver (that of the monodispersed approach) to implement the one- or two-groups interfacial area transport equations. In the first case 12 equations need to be solved:

 Momentum of the liquid phase: 3  Momentum of the gas phase: 3  Pressure: 1

 Mass of liquid: 1  Mass of gas: 1

 Turbulence of the liquid phase: 2  One-group interfacial area density: 1

In the second case 17 equations are considered. This is due to the fact that the second bubble groups is not sharing the same velocity field with the first one:

 Momentum of the liquid phase: 3  Momentum of the first gas phase: 3  Momentum of the second gas phase: 3  Pressure: 1

 Mass of the liquid phase: 1  Mass of the first gas phase: 1  Mass of the second gas phase: 1  Turbulence of the liquid phase: 2  One-group interfacial area density: 1  Two-group interfacial area density: 1

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8.3

Additional Variable for the Interfacial Area Density