Evaporation and Condensation Solution Procedure

Thermal Limited On

Below the saturation temperature (also referred to as boiling temperature), evaporation and condensation rates are controlled by the following:

Hydrodynamic effects - vapor is delivered to or carried away from the surface, which happens at a finite rate.
Thermal effects - heat is conducted towards or away from the interface to balance the latent heat of evaporation.

Simcenter STAR-CCM+ attempts to satisfy Eqn. (2793), in which ${\dot{Q}}_{v}$ is given by Eqn. (2794), which in turn relies on Eqn. (2792), which requires Eqn. (2774). Despite the apparent complexity, this condition is simply finding the interfacial temperature $T_{s}$ such that Eqn. (2793) is satisfied.

Saturation can be detected by either (or both simultaneously) of these conditions:

Figure 1. EQUATION_DISPLAY

\sum_{i}^{N_{V}} Y_{j} = 1

(2796)

Figure 2. EQUATION_DISPLAY

\sum_{i}^{N_{V}} Y_{\infty_{j}} = 1

(2797)

where $Y$ is the mass fraction.

The first condition (Eqn. (2796)) states that the temperature at the interface has reached the boiling temperature. The second condition (Eqn. (2797)) states that a pure vapor (without the presence of non-condensable gases) can only be in quasi-steady equilibrium at the interface if its temperature is the saturation temperature. If either of these conditions are detected, the hydrodynamic procedure above is cut short and the interfacial temperature is set to $T_{s a t}$ .

Thermal Limited Off

Without thermal limitation the evaporation rate is exclusively computed based on the mass flux considerations by taking only the hydrodynamic effects into account. It is assumed that the necessary heat flux can be transported towards or away the interface. Simcenter STAR-CCM+ first solves for the total evaporation rate Eqn. (2774) and component vise rates Eqn. (2792) are solved thereafter.

Final Application of Thermal Equilibrium

Once $T_{s}$ is found (either below the boiling temperature for hydrodynamically limited conditions or exactly at the boiling temperature for thermally limited conditions), Simcenter STAR-CCM+ finalizes the calculation by using the following steps:

Compute ${\dot{Q}}_{v}$ via enforcing the thermal equilibrium of Eqn. (2793).
Compute ${\dot{m}}_{v}$ using Eqn. (2795).
Compute ${\dot{m}}_{v, i}$ using Eqn. (2792).

Note that if $T_{s} < T_{s a t}$ , these final steps are not necessary: they are fully consistent with the quantities that are computed during the iterative process, and would therefore be already available. However, these final steps are presented to illustrate that, once $T_{s}$ has been found, evaporation rates can be calculated in an identical fashion for both the non-saturated and the saturated regimes.

Steam Condensation at a Dry Wall

The method that is outlined above runs into issues at perfectly dry cold walls that are in contact with a pure vapor (for example, at the start of the simulation).

The heat transfer coefficient in the film is generally given by:

Figure 3. EQUATION_DISPLAY

h_{f} = \frac{2 k_{f}}{h}

(2798)

so the following approximation (based on the assumption of piecewise linear temperature profile) holds:

Figure 4. EQUATION_DISPLAY

k_{f} \frac{d T}{d y} |_{f} \approx \frac{2 k_{f}}{h} (T_{f, c} - T_{s})

(2799)

where $T_{f, c}$ is the film liquid temperature at the cell center. For vanishing film thickness and $T_{s} > T_{f, c}$ , Eqn. (2799) tends to infinity, expressing that at zero film thickness the heat is carried towards the wall infinitely fast. At saturation conditions, the thermally limited model (which is assumed here) effectively predicts infinite condensation rates, which causes the simulation to diverge. (The heat transfer coefficient that is calculated by Simcenter STAR-CCM+ does not actually go to infinity, as a certain minimal thickness (currently 10^-8m) is used. However, the calculated value is still extremely high.)

Condensation at a dry wall does not happen in the form of a flat film, but rather as droplets growing at nuclei. The growth of these droplets is restricted by surface tension effects, and the interfacial contact area between vapor and liquid is small, so it is likely that these become limiting factors at the early stages of condensation.

To implement a model that behaves in the same way as the purely hydrodynamically limited model (that is, models the condensation as a fully-wetting film), and remains stable, a simplified drop-wise approach is used. The impact of the model is only on the interfacial area.

Consider a film cell with area $A$ and thickness $h$ . Assume that there is a nucleation density of $N$ (m^-2) and that the nuclei have a minimum radius of $R_{\min}$ . It is further assumed that the droplets form a 90 degree contact angle, and that they are all identical with a radius $R$ . For a given film thickness, the radius of the droplets (assuming hemispheres) can be expressed as:

Figure 5. EQUATION_DISPLAY

R = {(\frac{3 h}{2 π N})}^{1 / 3}

(2800)

It is assumed that the evaporation rates as calculated before still hold and that the only effect is due to the smaller total interfacial area between the vapor and the film. This ratio can be described as:

Figure 6. EQUATION_DISPLAY

f_{A} = 2 π N R_{e f f}^{2}

(2801)

$R_{e f f}$ is evaluated as $\max (R, R_{\min})$ where $R$ is determined through Eqn. (2800) and $R_{\min}$ is the Minimum Nuclei Radius that is specified as a property of the evaporation and condensation model.

This expression behaves as $N^{1 / 3} h^{2 / 3}$ . When the factor $f_{A}$ is smaller than 1, it is used as a multiplier for the evaporation rates. The value $N$ is specified as the Nucleation Density property of the evaporation and condensation model. For more information, see Evaporation-Condensation Model Properties.

This model stabilizes the solution without affecting the longer-term result, that is, once a fully-wetting film has formed and grown.