Evaporation and Condensation

Evaporation is a form of vaporization, the phase change from liquid to gas, which occurs when liquid molecules or atoms with high enough kinetic energy escape through the interface between the two phases. Condensation is the reverse process of vaporization and is the phase change from gas to liquid.

In Simcenter STAR-CCM+, evaporation and condensation deal with evaporation (without boiling) and condensation taking place at the free surface interface between a liquid and a gas phase. Both phases are modeled as multi-component mixtures that can include non-interacting (inert) species [606], [605]. Evaporation and condensation are treated as hydrodynamically limited, that is, the phases are considered to be in equilibrium at the interface and the driving force is the diffusion of species. Raoult’s law describes the phase equilibrium.

The evaporation rate (kg/s m²) for a particular component $i$ is given by:

Figure 1. EQUATION_DISPLAY

{\dot{m}}_{i} = - \frac{ρ_{g} D_{g, i} \frac{\partial Y_{g, i}}{\partial n} |_{s}}{1 - \sum_{j = 1}^{N_{v}} Y_{g, j}^{s}}

(2660)

where:

$ρ_{g}$ is the density of the gas phase
$Y^{s}$ is the component mass fraction at the surface
$D_{g, i}$ is the diffusion coefficient
$N_{v}$ is the total number of components undergoing evaporation or condensation
Subscript $g$ denotes a gas phase variable

The vapor pressure $p_{i}$ of a component is:

Figure 2. EQUATION_DISPLAY

p_{i} = a_{i} {p_{i}}^{*}

(2661)

where ${p_{i}}^{*}$ is the vapor pressure of the pure component $i$ and $a_{i}$ is the activity that depends on the liquid mixture. This expression can be written as:

Figure 3. EQUATION_DISPLAY

p_{i} = γ_{i} X_{L, i} {p_{i}}^{*}

(2662)

where $γ_{i}$ is the activity coefficient that accounts for deviations from ideal behavior of a mixture.

Raoult's law states that $γ_{i} \approx 1$ , an approximation that is used here. Given the vapor pressures of the components, their molar fractions at the interface can be derived from:

Figure 4. EQUATION_DISPLAY

X_{g, i}^{s} = \frac{p_{i}}{p}

(2663)

where $p$ is the pressure. The conversion to interface vapor mass fraction is done through:

Figure 5. EQUATION_DISPLAY

Y_{g, i}^{s} = \frac{X_{i}^{s} W_{i}}{\sum_{j = 1}^{N_{v}} X_{j}^{s} W_{j} + \sum_{j = 1}^{N_{g, p}} X_{j}^{s} W_{j}}

(2664)

where $N_{g, p}$ is the number of inert components in the gas phase. While the molar fraction of the evaporation components is known, it is unknown for the inert species (which do not undergo evaporation or condensation). A good approximation is the introduction of a background molar weight calculated a short distance away from the interface:

Figure 6. EQUATION_DISPLAY

W_{b g} = \frac{\sum_{j = 1}^{N_{g, p}} X_{j} W_{j}}{\sum_{j = 1}^{N_{g, p}} X_{j}}

(2665)

The interfacial background molar fraction is:

Figure 7. EQUATION_DISPLAY

X_{b g}^{s} = 1 - \sum_{j = 1}^{N_{v}} X_{j}^{s}

(2666)

Therefore the interfacial vapor mass fraction can be approximated as:

Figure 8. EQUATION_DISPLAY

Y_{g, i}^{s} \approx \frac{X_{i}^{s} W_{i}}{\sum_{j = 1}^{N_{v}} X_{j}^{s} W_{j} + X_{b g}^{s} W_{b g}}

(2667)

The numerical solution of Eqn. (2660) begins from the definition of the following scalar function $Y_{g, m}$ , for every interacting component:

Figure 9. EQUATION_DISPLAY

Y_{g, m} = α_{l} Y_{g, i}^{s} + α_{g} Y_{g, i}

(2668)

where $α_{l}$ is the liquid volume fraction and $α_{g}$ is the gas volume fraction. The approximated evaporation rate ${\dot{M}}^{'}_{i, c}$ for one cell can be obtained as:

Figure 10. EQUATION_DISPLAY

{\dot{M}}^{'}_{i, c} \approx - \frac{ρ_{g} D_{g, i} \nabla Y_{g, m} \nabla α_{l} V_{c}}{1 - \sum_{j = 1}^{N_{v}} Y_{g, j}^{s}}

(2669)

using the fact that the integral of $\nabla α_{l}$ , the gradient of liquid fraction, over a certain volume expresses the area of the interface it contains [580], [591].

Comparing Eqn. (2660) and Eqn. (2669), it is seen that $\nabla Y_{g, m}$ expresses the effect of $\nabla Y_{g, i}$ . A lowest order analysis shows that the relative error of the evaporation rate, in quasi-steady state, behaves as:

Figure 11. EQUATION_DISPLAY

ε_{r e l} \approx \frac{1}{H | \nabla α |}

(2670)

where $H$ expresses the thickness of the boundary layer above the interface.

As such, the accuracy depends on how well the location of the interface is determined in comparison with the boundary layer. When the interface is not smeared, ${| \nabla α |}^{- 1}$ , the reciprocal of the magnitude of the gradient of volume fraction, is a measure of the mesh size. This condition highlights the need for a sufficiently fine grid, to model the transport of species away from and towards the interface accurately. Similar considerations are valid for gradients of the temperature and pressure near the interface.

The sources for species, phases, and continuity are, based on Eqn. (2669), distributed over those cells that make up the interface, that is with non-vanishing gradient of the volume fraction, as shown in the above graph depicting the typical behavior of $α_{l}$ (solid line) and $- | \nabla α_{l} |$ (dashed line) at a free surface interface, along the surface normal coordinate $n$ .