### 5.13.6. The Droplet Condensation Model

The Droplet Condensation Model is useful for situations where a dry (or near-saturation) two-phase flow undergoes rapid pressure reduction leading to nucleation and subsequent droplet condensation. It is also useful to model additional condensation when droplets are already present in significant quantities. Typical applications include low-pressure steam turbines, in which context this model is also referred to as the Nonequilibrium Steam (NES) model. Such flows are typically transonic. The droplet phase can enter through the inlet or appear through various nucleation mechanisms, including homogeneous (volumetric) and heterogeneous (surface) nucleation. Presently, only one homogeneous nucleation model is available for selection based on Classical theory. Heterogeneous and alternate homogeneous nucleation models can be supplied using the available user-defined option.

The Droplet Condensation Model differs from the Thermal Phase Change model (see The Thermal Phase Change Model) in that the droplet diameter is calculated as part of the model rather than as a user input. This leads to improved accuracy. In order to do this a transport equation for droplet number must also be solved, which has as its source term nucleation contributions. This model differs from the Equilibrium Phase Change model (see Equilibrium Phase Change Model in the CFX-Solver Modeling Guide) in that it does not assume the flow to instantaneously reach equilibrium conditions, and therefore implicitly includes losses due to thermodynamic irreversibility.

In the following discussion, we consider a homogeneous multiphase system, in which the droplets move with the same velocity as the continuous phase. However, the model has been generalized to inhomogeneous systems as well.

The system of equations involves one continuous phase and any number of dispersed (condensed) phases. The condensed phases travel at the speed of the continuous phase. Any combination of condensed phases can exist in the solution so that for the continuous phase, mass conservation becomes:

| (5–268)

where the mass sources are summed over the condensed phases. The condensed phases can change size by condensation or evaporation.

For a condensed phase, mass conservation is:

| (5–269)

where each dispersed phase has a corresponding number equation of the form:

| (5–270)

and is the nucleation model with units defined as the number of droplets generated per unit time per unit volume of vapor and is the nucleated droplet mass based on the critical radius . Note that the droplets are transported with the mixture velocity because no slip is assumed between the phases. The usual constraint applies for the volume fractions where:

| (5–271)

In addition, global continuity and momentum equations are also solved as described in Homogeneous Hydrodynamic Equations. The continuous phase energy equation, in total enthalpy form, is

| (5–272)

where a similar energy transport equation can be applied for the dispersed phases, either in the context of small or large droplets as will be discussed subsequently. The Droplet Condensation model can be used for both small and large droplets. However, small and large droplets use different models for heat transfer and phase change. For large droplets, the heat transfer and phase change models described by the Thermal Phase Change model should be used. For small droplets (less than 1 ) the Small Droplet heat transfer model is appropriate; it sets the droplet temperature to

| (5–273)

where refers to the saturation temperature, refers to the supercooling level in gas phase, is the droplet radius, and is the critical radius at formation of the dispersed phase.

For small droplets, the interphase heat and mass transfer models are also modified to include the influence of the Knudsen () number on the Nusselt number. The dependence is required because droplet sizes vary significantly from the initial nucleated radius (in a non-continuum regime) in the range of angstroms. The droplet growth rate is:

| (5–274)

which is subsequently used to compute the interphase mass transfer rate in conjunction with an interfacial area density to be described later in this section.

The source of droplets into the domain is based on a nucleation model, which for classical nucleation models has the form of:

| (5–275)

where is a constant determined by the particular nucleation model, is the Gibbs free energy change at the critical radius conditions, is Boltzmann's constant, and is the supercooled vapor temperature. To compute the Gibbs free energy change a property database must be used that evaluates supercooled state properties. This requires an equation of state for the vapor phase amenable to extrapolation into regions within the saturation zone. The IAPWS and Redlich Kwong equations of state satisfy this requirement. A user-defined nucleation model can also be applied, enabling different homogeneous or heterogeneous nucleation models to be employed.

The interaction between the phases by mass transfer depends on calculating the droplet diameter, which, if a monodispersed distribution is assumed for droplets with a common origin, can be determined from the droplet number, . The relevant equation is then:

| (5–276)

with an interfacial area density defined as:

| (5–277)

The interfacial mass transfer term can then be computed with the known droplet growth rate and the interfacial area density:

| (5–278)

which can be used to obtain the heat transfer term:

| (5–279)

where is an upwinded total enthalpy, its value either for the continuous or dispersed phase depending on the direction of interphase mass transfer. In addition, is the heat transfer (per unit area) between the dispersed and continuous phase based on:

| (5–280)

Since the droplets in condensing systems are generally quite small (less than 1 ), it is assumed that the droplet temperature is uniform (a zero resistance model between the droplet surface and its internal temperature). This implies that almost all of the heat transfer either comes from the continuous phase during evaporation or goes into it during condensation.

The Nusselt () number underlying Equation 5–274 and Equation 5–280 is corrected to account for droplet sizes that span a wide Knudsen number () range (from free-molecular to continuum). The Nusselt number applied is:

| (5–281)

where is an empirical factor set to 3.18.

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