The following is assumed in the phase change model:

The mass generation rate in the classical nucleation theory during the nonequilibrium condensation process is given by the sum of mass increase due to nucleation (the formation of critically sized droplets) and also due to growth/demise of these droplets [269].

Therefore, is written as:

(14–542)

where is the average radius of the droplet, and is the Kelvin-Helmholtz critical droplet radius, above which the droplet will grow and below which the droplet will evaporate. An expression for is given by [727].

(14–543)

where is the liquid surface tension evaluated at temperature , is the condensed liquid density (also evaluated at temperature ), and is the super saturation ratio defined as the ratio of vapor pressure to the equilibrium saturation pressure:

(14–544)

The expansion process is usually very rapid. Therefore, when the state path crosses the saturated-vapor line, the process will depart from equilibrium, and the supersaturation ratio can take on values greater than one.

The condensation process involves two mechanisms, the transfer of mass from the vapor to the droplets and the transfer of heat from the droplets to the vapor in the form of latent heat. This energy transfer relation was presented by Hill in [246] and used in [269] and can be written as:

(14–545)

where is the droplet temperature, is the specific enthalpy of evaporation at pressure , is the vapor isobaric specific heat capacity, is the ratio of specific heat capacities, and is the specific gas constant for gaseous mixture of air and vapor.

Hill's droplet growth formula (Equation 14–545) is shown to predict reasonably well the Wilson Point pressure rise for nozzle flows. However, it tends to underestimate average droplet size distribution. To improve the average droplet size, Young's droplet growth formula is introduced from [725]. This formula is tunable with two modeling parameters, and :

(14–546)

where

and is the vapor subcooling:

with being the saturation temperature at pressure .

Other variables are as follows:

is the Knudsen number, is the mean free path of a vapor molecule, is the vapor dynamic viscosity, is the vapor thermal conductivity, is the vapor Prandtl number, is the evaporation coefficient, is a modeling parameter with default value 9. It is the growth coefficient that relates the condensation coefficient with the evaporation coefficient. is a modeling parameter with default 1. It is a coefficient that adjusts the distance from the droplet at which continuum processes, as opposed to free-molecular processes, occur, with typical values between 0 and 2.

Both formulas for the droplet growth are available in Fluent, with Young’s formula being the default.

The classical homogeneous nucleation theory describes the formation of a liquid-phase in the form of droplets from a supersaturated phase in the absence of impurities or foreign particles. The nucleation rate described by the steady-state classical homogeneous nucleation theory [727] and corrected for non-isothermal effects, is given by:

(14–547)

where is evaporation coefficient, is the Boltzmann constant, is mass of one molecule, is the liquid surface tension, and is the liquid density at temperature .

A non-isothermal correction factor, , is given by:

(14–548)

where is the specific enthalpy of evaporation at pressure and is the ratio of specific heat capacities.