Non-Equilibrium Condensation

Mass Transfer

Mass transfer due to condensation occurs when the system departs from the equilibrium condition.

While, in principle, condensation starts when the saturation line is crossed, this is not the case in applications such as transonic/supersonic nozzle flows. Due to the expansion and associated rapid cooling, the vapor phase can depart from the equilibrium distribution into a metastable state and reach a large degree of supersaturation before condensation actually occurs.

For homogeneous nucleation to occur, the vapor phase needs to develop a supercooling level within the range of 30-40 K. The growth of the droplets, thereafter, leads to a significant release of latent heat in the flow. As a consequence, both droplet and vapor temperatures are increasing and have a strong impact on the flow pattern. As the flow becomes thermally choked due to the heat's compressive effects, a steady condensation shock forms in the nucleation zone, interrupting the nucleation process.

Supersaturation / Supercooling

In order for a liquid droplet to form from the vapor phase, surface free energy needs to be supplied and bulk free energy decreased. For spherical droplets, this change in (Gibb's [486]) free energy can be expressed by:

$$\mathrm{\Delta }G=4\pi r^2\sigma -\frac{4}{3}\pi r^3{\rho }_{d}R{T}_{v}ln\left(S\right)$$

(2861)

where:

• $r$ is the droplet radius.
• $T_v$ is the vapor temperature.
• $\sigma$ is the surface tension.
• $R$ is the gas constant.
• $S$ is the supersaturation ratio, defined as

$$S=\frac{P_{abs}}{P_{sat}{(T}_v)}$$

(2862)

See [486] for $\mathrm{\Delta }G$ plotted against the droplet radius r at different supersaturation ratios. In brief, under supersaturation conditions $S>1$ , the first term in Eqn. (2861) (the free energy increase due to the liquid bulk surface formation) is positive and the second term (the increase in free energy due to the interphase formation between vapor and liquid) is negative.

Overall, $\mathrm{\Delta }G$ increases with radius r. However, at $S>1$ and at critical radius $r^{*}$ also known as Kelvin-Helmholtz radius, $\mathrm{\Delta }G$ attains a maximum value $\mathrm{\Delta }{G}^{*}$ . From that point, a droplet with $r\ge r^{*}$ will reduce the free energy of the system and therefore further growth of the droplet size becomes energetically favorable. The critical radius $r^{*}$ is defined as:

$$r^{*}=\frac{2\sigma }{{\rho }_{l}R{T}_v\ln \left(S\right)}$$

When supersaturation ratio increases, the free energy barrier ${\mathrm{\Delta }G}^{*}$ and the critical cluster size $r^{*}$ decrease. This suggests that the probability that the barrier ${\mathrm{\Delta }G}^{*}$ is overcome and nucleation is triggered, increases with the level of supersaturation and hence, supercooling. With the process of homogeneous nucleation starting, the phase transition will be initiated.

(2863)

Classical Nucleation Theory

At low or moderate supersaturation ratios $S$ , the nucleation rate $J$ , is negligibly small. However, when the supercooling temperature $\mathrm{\Delta }T=T_{\text{sat}}-T_v$ , increases the nucleation rate also increases rapidly. The nucleation rate, in the sense of homogeneous nucleation, is formulated via the classical nucleation theory. The classical nucleation theory defines the isothermal, homogeneous nucleation rate $J_{CL}$ as the level of super-saturation that the vapor phase has to overcome to initiate the transitions:

$$J_{CL}=q_{c}S\frac{{\rho }_v^2}{{\rho }_l}\sqrt{\frac{2\sigma }{\pi m^3}}\exp (-\frac{4\pi r^{*}\sigma }{3k_B{T}_v})$$

(2864)

where:

• $q_c$ is a dimensionless condensation coefficient.
• ${\rho }_v$ is the vapor density.
• ${\rho }_l$ is the liquid density.
• $m$ is the molecular mass.
• $k_B$ is the Boltzmann constant.

Further developments have been made to the classical nucleation theory, which have led to several adjustments of $J_{CL}$ , (see [526] for detailed description).

The resulting nucleation rate is formulated as:

$$J=q_{c}C\frac{{\rho }_v^2}{{\rho }_l}\sqrt{\frac{2\sigma }{\pi m^3}}\exp (-\frac{4\pi r^{*}\sigma }{3k_B{T}_v})$$

(2865)

where C is Kantrowitz' correction factor that accounts for the non-isothermal effects, originally not included in the classical theory. This correction factor is defined as:

$$C=\frac{1}{1+2\frac{\gamma -1}{\gamma +1}\frac{L}{R{T}_v}(\frac{L}{R{T}_v}-\frac{1}{2})}$$

(2866)

where:

• $L$ is the vaporization latent heat.
• $\gamma$ is vapor specific heat ratio.

注

The classical nucleation theory is stated in terms of steady-state assumptions. However, because the characteristic time-scale of the nucleation process is significantly lower than that of both temperature and flow field changes for such typical applications, it can usually be applied to both steady and unsteady flows.

Droplet Growth

The growth of a droplet of critical size is determined by the mass flux of supercooled vapor molecules towards the droplet and the latent heat release towards the vapor. The transfer of latent heat causes an instant increase of the vapor temperature. As a consequence, the vapor pressure will increase during this process, also commonly referred to as a condensation shock. Gerber [465] defines the droplet growth mass transfer from the heat flow balance at the droplet surface:

where:

• $q_v$ is the vapor heat flux towards the interface.
• $q_d$ is the droplet heat flux towards the interface. In case of very small droplets (below 1 micron), the internal temperature gradients are assumed insignificant, and this property is then neglected in the droplet growth mass transfer formulation.
• ${\beta }_d$ is the interaction area density, defined as:

  $${\beta }_d=\frac{3{\alpha }_d}{r}$$

  (2868)

  where ${\alpha }_d$ is the droplet volume fraction.

• $h_v$ is the vapor enthalpy evaluated at the corresponding temperature $T_v$ .
• $h_d$ is the droplet enthalpy evaluated at the corresponding temperature $T_d$ . For small droplets, droplet temperature $T_d$ is expressed as a droplet size relation:

  $$T_d=T_{sat}-\mathrm{\Delta }T\frac{r^{*}}{r}$$

  (2869)

In case of small droplets, $q_d$ is neglected, and the condensation is accounted for by the vapor heat flux the droplet surface $q_v$ , which is expressed in terms of the Nusselt number ${Nu}_v$ :

where ${\lambda }_v$ is the vapor thermal conductivity.

The Nusselt number can be specified using the following methods:

Gyarmathy

Gyarmathy's droplet growth law [471] defines the Nusselt number as:

$${Nu}_v=\frac{2}{1+3.18Kn}$$

(2871)

$Kn=\frac{l}{2r}$ is the Knudsen number, where $l$ is the mean, free path of the vapor molecules, given as:

$$l=\frac{3{\mu }_v}{P_{abs}}\sqrt{\frac{\pi R{T}_v}{8}}$$

(2872)

where ${\mu }_v$ is the dynamic viscosity of vapor.

For very small Knudsen numbers, the continuum hypothesis holds and the transfer process is mainly governed by diffusion. For large Knudsen numbers, the process is governed by kinetic gas theory that is, impingement of vapor molecules onto the droplets.

Young

Young's droplet growth law [574] defines the Nusselt number as:

$${Nu}_v=\frac{2}{\frac{1}{1+2\beta Kn}+3.78(1-\nu )\frac{Kn}{\mathrm{Pr}}}$$

(2873)

where, $\mathrm{Pr}=\frac{C_p{\mu }_v}{{\lambda }_v}$ is the Prandtl number, and

$$\nu =\frac{R{T}_{sat}}{L}\sqrt{(\alpha -0.5-\frac{2-q_c}{2q_c}\frac{\gamma +1}{2\gamma }\frac{C_p{T}_{sat}}{L}}$$

(2874)

where:

• $C_p$ is the specific heat.
• $\gamma =\frac{C_p}{C_v}$ is specific heat ratio of the vapor phase.
• $\alpha$ , and $\beta$ are user-specified dimensionless coefficients.