In the homogeneous equilibrium model (HEM) one assumes that the velocity, temperature and pressure between the phases or components are equal. This assumption is based on the belief that differences in these three potential variables (and chemical potential if chemical reactions are considered) will promote momentum, energy, and mass transfer between the phases rapidly enough so that equilibrium is reached. For example, when one phase is finely dispersed in another phase generating large interfacial area, under certain circumstances this assumption can be made; e.g., bubbly flow of air in water or steam in water at high pressures. The resulting equations resemble those for a pseudo-fluid with mixutre properties and an equation of state which links the phases to obtain these mixture thermodynamic properties. Whenever the HEM model is used it is advisable to check the validity of the equilibrium assumptions by using more accurate theoretical models for comparison. For example, rapid acceleration or pressure changes cannot be always accurately modelled with the HEM model; i.e., discharge of flashing vapor-liquid mixtures, or shock wave propagation through a multiphase medium. This is especially true when the pressure change is large when compared to the ambient pressure, or any of the driving potentials are large relative to their reference values. Such a 'rule-of-thumb' is very crude and one must carefully consider the timescales for equilibration of these driving potentials with allowable characteristic times for the problem of interest.
The governing equations for the HEM model are presented in Table 3.2 where the geometry has been chosen to be a one-dimensional channel inclined from the horizontal by a known angle, , (Figure 3.1). The microscopic equations have been averaged over the channel cross-sectional area using the techniques first proposed by Ishii (1974), leaving a partial differential equation in time, t, and the axial space dimension, z. The definitions of the mixture thermodynamic properties (e.g., , u, v) consider only two-phases but can be simply extended to more phases or components. The extension to more than one-dimension is not as straight forward and one is referred to the general formulation of Bird, et al. (1960). Note that the inclusion of the mixture thermodynamic properties can be followed from Table 3.2.
The multiphase transport properties of viscosity and thermal conductivity ( , k) are another matter, because it is not clear how one should average their effect in an area-average, mass average or volume-average sense. In many situations such as for pressure drop calculations the mixture transport properties have been arbitrarily averaged on a volume average or mass average basis, e.g.,
However, these averaging schemes are not exact and are usually empirically corrected by fitting coefficients to a set of experimental data. In other situations the effect of multiphases are neglected and the liquid or gas property values for viscosity on the thermal conductivity are used, e.g., when the amount of liquid in the channel is large (low quality or void fraction), the viscosity can be taken to be that of the liquid.