# Homogeneous Flow Model for Two-Phase Flow

### From Thermal-FluidsPedia

The central assumption of the homogeneous flow model is that the two phases travel at equal velocities and mix well; therefore, they can be treated as if there is only one phase. This model works better for two-phase flow near the critical point, where the differences between the properties of the liquid and vapor are insignificant, or when the mass velocity of the two-phase flow is very high so that the flow regime is either bubbly or misty flow. In arriving at the homogeneous model for two-phase flow, area averaging is performed for both phases and the resultant governing equations become one-dimensional in nature. The density of the homogeneous mixture satisfies the relation

i.e.,

The mass flow rates of the liquid and vapor phases defined as (see Concepts and Notations for Two-Phase Flow)

Substituting eqs. (3) and (4) into the from definition of void fraction

yields

Considering the definition of quality:

the void fraction becomes

The total mass flux in the channel becomes

The governing equations for the homogeneous model include continuity, momentum and energy equations, which are listed below:

where the angle brackets, “,” which represent the area average, have been dropped for ease of notation. *P* is perimeter, *p* is pressure, *q*''_{w} is heat flux at the wall, and *q*''' is internal heat generation per unit volume within the fluid.

Substituting eq. (7) into eq. (10), the energy equation becomes

For steady-state two-phase flow in a circular tube with constant cross-sectional area, the momentum eq. (9) reduces to

The three terms on the right-hand side of eq. (12) represent pressure drops due to friction, *d**p*_{F} / *d**z*, acceleration, *d**p*_{a} / *d**z*, and gravity, *d**p*_{g} / *d**z*. Thus, eq. (12) can be rewritten as

where the pressure drop due to friction, *d**p*_{F} / *d**z*, must be obtained using appropriate correlations.

The energy equation (including mechanical and thermal energy) for steady-state two-phase flow in a circular tube with constant cross-sectional area can be obtained by simplifying eq. (11),

The enthalpy of the two-phase mixture can be expressed as For a two-phase flow system with condensation or evaporation, where kinetic and potential energy as well as internal heat generation can be neglected, eq. (14) reduces to

## References

Faghri, A., and Zhang, Y., 2006, *Transport Phenomena in Multiphase Systems*, Elsevier, Burlington, MA.