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23.6.3 Phase Change Model

The following is assumed in the phase change model:

The mass generation rate $\Gamma$ 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 [ 153].

Therefore, $\Gamma$ is written as:

 \Gamma = \frac{4}{3} \pi \rho_{\rm l} I {r_{\rm *}}^3 + 4 \p... ...} \eta \overline{r}^2 \frac{\partial \overline{r}}{\partial t} (23.6-7)

where $\overline{r}$ is the average radius of the droplet, and $r_{\rm *}$ is the Kelvin-Helmholtz critical droplet radius, above which the droplet will grow and below which the droplet will evaporate. An expression for $r_{\rm *}$ is given by [ 411].


 r_{\rm *} = \frac{2 \sigma}{\rho_{\rm l} R T \ln{S}} (23.6-8)

where $\sigma$ is the liquid surface tension evaluated at temperature $T$, $\rho_{\rm l}$ is the condensed liquid density (also evaluated at temperature $T$), and $S$ is the super saturation ratio defined as the ratio of vapor pressure to the equilibrium saturation pressure:


 s = \frac{P}{P_{\rm sat} (T)} (23.6-9)

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 $S$ 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 in [ 409] and used in [ 153] and can be written as:


 \frac{\partial \overline{r}}{\partial t} = \frac{P}{h_{\rm l... ... \pi R T}} \frac{\gamma + 1}{2 \gamma} C_{\rm p}(T_{\rm0} - T) (23.6-10)

where $T_{\rm0}$ is the droplet temperature.

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 [ 411] and corrected for nonisothermal effects, is given by:


 I = \frac{q_{\rm c}}{(1+\theta)}{\left ( \frac{\rho_{\rm v}^... ...t ( \frac{4 \pi {r_{\rm *}}^2 \sigma}{3 K_{\rm b} T} \right )} (23.6-11)

where $q_{\rm c}$ is evaporation coefficient, $k_{\rm b}$ is the Boltzmann constant, $M_{\rm m}$ is mass of one molecule, $\sigma$ is the liquid surface tension, and $\rho_{\rm l}$ is the liquid density at temperature $T$.

A nonisothermal correction factor, $\theta$, is given by:


 \theta = \frac{2(\gamma - 1)}{(\gamma + 1)} \left ( \frac{h_... ...lv}}{RT} \right ) \left ( \frac{h_{\rm lv}}{RT} - 0.5 \right ) (23.6-12)

where $h_{\rm lv}$ is the specific enthalpy of evaporation at pressure $p$ and $\gamma$ is the ratio of specific heat capacities.


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