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Two-phase flow modeling

In order to derive a two-phase flow model, ie equations which describe the conservation of mass, momentum and energy, one can pursue a least three different ways.

One way is to derive in a formal way the instantaneous field conservation equations, one set for each of the two phases. The resulting two-fluid model which also includes a set of interface conditions is the most general one, ie applicable to every two-phase flow regardless of, for instance, the fluid, flow pattern and channel geometry. Because of the very high complexity that two-phase flows exhibit these equations include quantities, especially in connection with the interface conditions, which at present cannot be modeled. The remedy is to sacrifice some of the generality and detailed information by making certain assumptions in favor of improved tractability. It is, for instance, possible to average the field equations over the cross-sectional area in order to obtain a one-dimensional model.

The second way, which we will call the engineering or phenomenological approach, is to derive a one-dimensional form of the conservation equations by considering a control volume which consists of a differential piece of the channel in the flow direction. This is the way most people derive the famous homogeneous and separated two-phase flow models. It is, however, important to recognize that it is very hard to establish directly the assumptions and restrictions of the models using this derivation--the conservation equations are, in fact, postulated! It is necessary to compare the resulting equations with equations derived by the formal way and thus originate from the field equations.

A third way is to divide the cross section of the channel into a number of communicating channels, the so-called subchannels. The conservation equations are then applied to each of these subchannels and to effect closure of the set of equations we have external relations which describe the transport in the transverse direction. Traditionally, the flow in the subchannels is treated as a mixture (mixture model) in which a slip between the phases exists. The main problem with these kinds of models is to obtain accurate expressions for the transverse transport. These models are however, as far as the author knows, the only ones which recognize the 3-D nature of two-phase flow.

Schematically, we can differentiate between the different two-phase flow models by the following criteria.

.

General model:

)

One-dimensional models (1-D).

)

Subchannel models (3-D).

.

Fluid model:

)

Homogeneous (no slip).

)

Mixture--Slip.

)

Mixture--Drift flux.

)

Two fluid.

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Thermodynamic equilibrium:

)

Full equilibrium, ${\mbox{h}}_\ell= {\mbox{h}}_{\ell,{\mbox{\protect\scriptsize sat}}}(p)$ and ${\mbox{h}}_g= {\mbox{h}}_{g,{\mbox{\protect\scriptsize sat}}}(p)$.

)

Partial equilibrium, ${\mbox{h}}_\ell= {\mbox{h}}_{\ell,{\mbox{\protect\scriptsize sat}}}(p)$ or ${\mbox{h}}_g= {\mbox{h}}_{g,{\mbox{\protect\scriptsize sat}}}(p)$.

)

Full non-equilibrium.

.

Fluid dynamics:

)

Steady-state.

)

Transient.

In the subsequent sections we derive a steady-state one-dimensional set of conservation equations which considers the fluid as a mixture. Before we consider the conservation equations we however have to introduce the Zuber-Findlay notation which is commonly used in the description of one-dimensional two-phase flows.

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