Boiling

Wall Boiling

When a liquid is in contact with a wall that is maintained at temperatures $T_{wall}$ above the saturation temperature of the liquid, boiling eventually occurs at that liquid-solid interface. In this case, boiling occurs in three characteristic stages:

• Nucleate boiling involves a creation and growth of vapor bubbles on a heated surface, which rise from discrete points on a surface. The temperature of the surface is only slightly above the saturation temperature of the liquid. In general, the number of nucleation sites increases with an increasing surface temperature. An increased surface roughness can create more nucleation sites, while an exceptionally smooth surface can result in superheating.
• Film boiling occurs when the critical heat flux is exceeded and a continuous vapor film covers the heated surface. The vapor layer has a lower thermal conductivity so the vapor layer typically “insulates” the surface.
• Transition boiling occurs at surface temperatures between the maximum attainable in nucleate and the minimum attainable in film boiling. It is an intermediate, unstable form of boiling with elements of both types.

There are two distinct choices for modeling boiling in Simcenter STAR-CCM+, namely the Rohsenow and the Transition Boiling models. The former uses the Rohsenow nucleate boiling model applicable for boiling at relatively low solid temperatures. A film boiling model is also integrated into the Rohsenow implementation in Simcenter STAR-CCM+ to facilitate its use with boiling at high solid temperatures. The Transition Boiling model has expressions for nucleate and transition boiling.

Rohsenow Boiling Model

The empirical correlation that Rohsenow [692] presented is used to calculate the surface heat flux due to boiling:

$$q_{bw}={{\mu }_{l}h}_{lat}\sqrt{\frac{g\left({\rho }_l-{\rho }_v\right)}{\sigma }}{\left(\frac{C_{p_l}\left(T_w-T_{sat}\right)}{{C_{qw}h}_{lat}{Pr}_l^{n_p}}\right)}^{3.03}$$

(2671)

In this equation, ${\mu }_l$ , $h_{lat}$ , $C_{p_l}$ , ${\rho }_l$ , and ${Pr}_l$ are the dynamic viscosity, latent heat, specific heat, density, and Prandtl number of the liquid phase, $n_p$ is the Prandtl number exponent (1.73 by default), $g$ is gravity, and ${\rho }_v$ is the vapor density, $\sigma$ is the surface tension coefficient at the liquid-vapor interface, $T_w$ is the wall temperature, $T_{sat}$ is the saturation temperature, and $C_{qw}$ is an empirical coefficient varying with the liquid-surface combination.

The vapor mass generation rate ${\dot{m}}_{ew}$ over the area that nucleation sites cover is:

$${\dot{m}}_{ew}=\frac{C_{ew}{q}_{bw}}{h_{lat}}$$

(2672)

where $C_{ew}$ is a model constant stating how much of the boiling heat flux is used for creation of vapor bubbles.

If the Rohsenow correlation is applied outside its range of applicability (for example, to the film boiling regime), unrealistically high heat fluxes could result. Fluid temperatures can become higher than the near wall temperature. This behavior stems from the fact that the Rohsenow correlation does not depend on the fluid temperature; heat enters the domain irrespective of the fluid temperature.

To prevent this condition, the heat flux that the correlation calculates is multiplied by:

$$max[0,min(\left(\frac{T_w-T}{Tw-T_{sat}}\right),1)]$$

(2673)

where $T$ is the fluid temperature near the heated wall. Thus, if $T<T_{sat}$ , the Rohsenow correlation is used directly, and if $T>T_w$ the boiling heat flux is zero. For fluid temperatures which are between the wall and the saturation temperature, only a fraction of the heat flux predicted by the correlation is used.

Film Boiling Model

At a sufficiently high wall temperature, a layer of vapor is created above the heated wall. As a result the liquid is no longer in contact with the heated wall. The evaporation occurs at the vapor liquid interface instead of at the heated wall. The vapor film that is created between the liquid and the heated wall acts as an insulator, which slows down the heat transfer process considerably. This process is known as film boiling.

The film boiling model assumes that the film thickness is resolved in one or more cells. That is, the vapor occupies one or more cells close to the heated wall. In this case, the expressions for the wall heat flux are the same as in the case of single-phase flows. If it is not possible to have a mesh which resolves the film thickness, you specify the value of the vapor volume fraction ${\alpha }_{filmBoiling}$ , which indicates when the film boiling starts. For values of the vapor volume fraction smaller than ${\alpha }_{filmBoiling}$ the nucleate boiling is assumed, and for larger values it is assumed that the film boiling takes place. If the film is resolved and the cell next to the heated wall is filled with vapor, the results do not depend on values of ${\alpha }_{filmBoiling}$ from 0 through 1.

The film-boiling vapor volume fraction ${\alpha }_{filmBoiling}$ is the value of volume fraction of the vapor phase which indicates full transition to film boiling.

If the VOF boiling model is active, you can specify ${\alpha }_{filmBoiling}$ for each wall boundary. A value of ${\alpha }_{filmBoiling}$ larger than 1 implies that there is no film boiling along these wall boundaries.

For values of volume fraction less than ${\alpha }_{filmBoiling}$ , a transition function is:

$$f\left({\alpha }_{vapor}\right)=1-\text{min}{(1,{\alpha }_{vapor}/{\alpha }_{filmBoiling})}^8$$

(2674)

This function controls the fraction of boiling at the wall that is considered to be nucleate boiling. Here ${\alpha }_{vapor}$ is the vapor volume fraction at the wall.

Transition Boiling Model

You can specify the constants of this boiling model. The equations are as follows:

$$q_{boiling}\left(\Delta T\right)=q_{\text{max}}{S\varphi \left(\frac{\Delta T}{\Delta T_1}\right)}^{K_1}0\le \Delta T\le \Delta T_1$$

(2675)

$$q_{boiling}\left(\Delta T\right)=q_{\text{max}}S\left[1-4\left(1-\varphi \right){\left(\frac{\Delta T-{\Delta T}_{\text{max}}}{\Delta T_2-\Delta T_1}\right)}^2\right]\Delta T_1\le \Delta T\le \Delta T_2$$

(2676)

$$q_{boiling}\left(\Delta T\right)=q_{\text{max}}{S\varphi \left(\frac{\Delta T}{\Delta T_2}\right)}^{{-K}_2}\Delta T_2\le \Delta T$$

(2677)

The first of the above equations simulates nucleate boiling, while the remaining two simulate transition I and transition II boiling. The three regimes are shown graphically in the following figure.

There are five empirical constants in the above model, namely, $K_1$ , $K_2$ , $\Delta T_1$ , $\Delta T_2$ , and $q_{\text{max}}$ , all of which are positive. $\Delta T_{max}$ is $\frac{1}{2}(\Delta T_1+\Delta T_2)$ .The constant $\varphi$ is set equal to 0.75. Further, $\Delta T_2$ must be greater than $\Delta T_1$ . (In the above expressions $K_2$ is positive even though it is a negative exponent because the sign is accounted for in the implementation.) The scale factor $S$ in the above expressions is not treated as a sixth constant. It allows you to scale $q_{\text{max}}$ up or down for a given boundary. All the constants are likely to be application-specific. Hence, it is recommended that default values for the constants be tuned with credible measurements. The default values are from fits of boiling experiments of Ellion [587] using an apparatus with a particular geometry.

As mentioned in the previous section, ${\alpha }_{filmBoiling}$ is the metric used to toggle between the nucleate and film boiling regimes in the Rohsenow Boiling Model. However, in the Transition Boiling Model, ${\alpha }_{filmBoiling}$ is not used; instead $\Delta T_1$ and $\Delta T_2$ are the metrics that switch between the three expressions (or regimes).

The total heat flux at the wall is the sum of the heat fluxes due to boiling, convection of the liquid-vapor mixture and thermal radiation. In the above equations, $q_{\text{max}}$ is the maximum of the boiling heat flux component; it is not the critical heat flux which is a maximum of the total heat flux. The point on the curve corresponding to the minimum of the total heat flux is the Leidenfrost point and it occurs in the Transition II boiling regime. At the Leidenfrost point, vapor completely covers the surface and heat transfer is primarily driven by mixture convection and radiation with a negligible boiling contribution.

In the CFD implementation of the above boiling model, the boiling heat flux contribution to the total heat flux is considered to be zero when there is no liquid to boil in any cell that is next to the wall. Further, if the fluid temperature $T$ is above the saturation temperature $T_{sat}$ the boiling heat flux is calculated according to Eqn. (2673).

Bulk Boiling

The heat transfer to the interface between the phases is used to compute the rate of evaporation or condensation ${\dot{m}}_{ec}$ . The following are assumed:

• The vapor bubbles are on the saturation temperature $T_{sat}$
• The temperature of the liquid $T_l$ is approximated by the temperature of the liquid-vapor mixture $T$
• All the heat flux from the liquid to the interface is used in mass transfer, that is, evaporation or condensation:

  $${\dot{m}}_{ec}=\frac{\text{HTCxArea}\left(T-T_{sat}\right)}{h_{lat}}$$

  (2678)

  where $h_{lat}$ is the latent heat of vaporization. $\text{HTCxArea}$ is the heat transfer coefficient between the vapor bubbles and the surrounding liquid, which is multiplied by the specific contact area (contact area per unit volume) between the two.