Wall Boiling

The wall boiling model for Mixture Multiphase (MMP) is aimed at forced-flow, subcooled, boiling applications. In subcooled boiling, local boiling occurs on a heated surface despite the mean temperature of the liquid being less than the saturation temperature. The degree of liquid subcooling is expressed either as a temperature difference or an enthalpy difference below the saturation value.

Provided that the flow regime near the wall remains close to bubbly flow, a standard subcooled boiling model can be pushed towards zero subcooling conditions (that is, saturated boiling). The model can also be pushed towards higher heat fluxes approaching departure from nucleate boiling (DNB) conditions. The mixture is in contact with the wall, and the convective heat flux of liquid and vapour are lumped together.

The subcooled boiling model of Kurul and Podowski [494] is used here. The model assumes that all the heat flux from the wall is transferred to the liquid next to the wall, and is then partitioned into three components. Adopting the notation ${{\dot{q}}^{"}}_w$ for heat flux, the partition of the wall heat flux could be written as:

These components of heat flux are described below:

• ${{\dot{q}}^{"}}_{conv}$ is the convective heat flux given which describes the removal of heat by the MMP mixture convection. For MMP wall boiling model, there one convection contributions from vapor and liquid, always the mixture in contact with the wall. This term is taken care of by the energy model.
• ${{\dot{q}}^{"}}_{evap}$ is the evaporative heat flux given by Eqn. (2918), which describes the power that is used to produce bubbles from nucleation to departure. This process is driven by the wall superheat, $T_w-T_{sat}$ . In fixed heat-flux applications, once boiling has started, this term is responsible for maintaining a wall temperature that is slightly higher than the saturation temperature.
• ${{\dot{q}}^{"}}_{quench}$ is the quenching heat flux given by Eqn. (2920), describes the enhancement of heat transfer, due to the replacement of a departing bubble by an influx of cooler liquid farther away from the wall. This process is dominated by the temperature difference between the wall and the liquid next to the wall. Bubble-induced quenching is also known in literature as liquid agitation or pumping. Quenching is less important when the liquid is close to saturation temperature, but in highly subcooled flows, $T_l«T_{sat}$ , it makes an important contribution.

Vapor Contribution

Subcooled boiling models start with the assumption that only the liquid phase is in contact with the wall. However, when the vapor volume fraction next to the wall becomes high enough, liquid access to the wall and its capability to remove heat from the wall are both restricted. This can occur unintentionally during a subcooled boiling simulation, as it passes through various intermediate states from initial conditions towards a converged solution.

Alternatively, the vapor contribution is relevant when studying the transition to DNB (Departure from Nucleate Boiling) conditions as wall heat flux is further increased.

When liquid access is restricted, vapor heat transfer removes some fraction of the wall heat flux and causes an increase in wall temperature. The heat fluxes can be represented as:

$${{\dot{q}}^{"}}_w={{\dot{q}}^{"}}_{conv}+{({\dot{q}}^{"}}_{evap}+{{\dot{q}}^{"}}_{quench})\left(1-K_{dry}\right)$$

(2917)

where:

• $K_{dry}$ is the wall dryout area fraction.

Evaporative Heat Flux

An evaporative heat flux is constructed for the following nucleate boiling mechanism:

• Bubbles nucleate and grow on suitably sized cavities in the wall.
• Bubbles reach a critical size, at which point the forces holding the bubble to the wall are no longer balanced.

The evaporative heat flux is:

$${{\dot{q}}^{"}}_{evap}=n″f\left(\frac{\pi d_w^3}{6}\right){\rho }_g{h}_{lg}$$

(2918)

where:

• $n″$ is the nucleation site number density
• $f$ is the bubble departure frequency
• $d_w$ is the bubble departure diameter
• ${\rho }_g$ is the vapor phase density
• $h_{lg}$ is the latent heat

The evaporative mass flux, ${{\dot{m}}^{"}}_{evap}$ , is the rate of conversion of liquid to vapor per unit wall area:

$${{\dot{m}}^{"}}_{evap}=\frac{{{\dot{q}}^{"}}_{evap}}{h_{lg}}$$

(2919)

Bubble Induced Quenching Heat Flux

When a bubble detaches from the wall, the space it occupied is replaced by cooler liquid. Quenching heat transfer is the component of heat flux that is used in heating this replacement liquid, as modeled by Del Valle and Kenning [447].

The bubble induced quenching heat flux is implemented using the temperature difference between the wall and the liquid next to the wall, $T_l$ , together with a slowly varying correction factor, $s_{quench}$ :

$${q″}_{quench}=h_{quench}{s}_{quench}\left(T_{wall}-T_l\right)$$

(2920)

The correction factor is defined as:

$$s_{quench}=\frac{T_{wall}-T_{quench}}{T_{wall}-T_l}=\frac{T^{\text{+}}\left(y_{quench}\right)}{T^{\text{+}}\left(y_c\right)}$$

(2921)

where:

• $h_{quench}$ is the quenching heat transfer coefficient
• $T_{wall}$ is the wall temperature
• $T_{quench}$ is the temperature at which liquid is brought to the wall by the departure of a bubble
• $T^{\text{+}}\left(y\right)$ is the dimensionless temperature profile that is formed by scaling wall heat flux to the liquid using liquid wall turbulence scales.
• $y_c$ is the distance from the wall to the nearest cell center.
• $y_{quench}$ is the representative distance from the wall at which cold liquid is drawn by the departure of a bubble. You specify this distance in terms of bubble departure diameters or of liquid wall turbulence scales.

Overall Energy Balance at the Wall

The overall energy balance at the wall for the combined models for subcooled boiling with possible dryout is as follows:

$${{\dot{q}}^{"}}_w={{\dot{q}}^{"}}_{wm}+{{\dot{q}}^{"}}_{w\lg }$$

(2922)

where:

• ${{\dot{q}}^{"}}_{wm}$ is the heat flux between the wall and the mixture:

$${{\dot{q}}^{"}}_{wm}={{\dot{q}}^{"}}_{conv}+\left(1-K_{dry}\right){{\dot{q}}^{"}}_{quench}$$

(2923)

• ${{\dot{q}}^{"}}_{w\lg }$ is the heat flux between the wall and liquid-gas interface:

$${{\dot{q}}^{"}}_{w\lg }=\left(1-K_{dry}\right){{\dot{q}}^{"}}_{evap}$$

(2924)