Wall Boiling

The subcooled boiling model of Kurul and Podowski [494], with contributions by many other workers, 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:

These components of heat flux are described below:

• ${{\dot{q}}^{"}}_{conv}$ is the convective heat flux given by Eqn. (2112), which describes the removal of heat by single-phase turbulent convection on those parts of the wall that are not affected by boiling. In applications with fixed wall heat flux, this term defines the point at which the wall first exceeds saturation temperature.
• ${{\dot{q}}^{"}}_{evap}$ is the evaporative heat flux given by Eqn. (2114), which describes the power that is used to produce bubbles from nucleation to departure. This term is a strong function of 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. (2116), which 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. 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:

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

(2111)

where:

• ${{\dot{q}}^{"}}_{dry}$ is the vapor contribution to convective heat flux (see Eqn. (2113)), based on single-phase turbulent convection by the vapor.
• $K_{dry}$ is the wall contact area fraction for the vapor (see Eqn. (2123)), based on either an expression for $K_{dry}$, or a transition volume fraction representing the start of wall dryout.

Convective Heat Flux

The turbulent convection fluxes are based on single-phase wall treatments.

For liquid contact with the wall, this flux is:

$${{\dot{q}}^{"}}_{conv}=\frac{{\rho }_l{c}_{pl}{u_l}^{*}}{t_l^{\text{+}}}\left(T_w-T_l\right)$$

(2112)

For vapor contact with the wall, this flux is:

$${{\dot{q}}^{"}}_{dry}=\frac{{\rho }_g{c}_{pg}{u_g}^{*}}{t_g^{\text{+}}}\left(T_w-T_g\right)$$

(2113)

The volume fraction does not modify the convection contributions as the contributions are in non-boiling heat transfer, where volume fraction represents the wall contact area fraction for each phase (compare this with Eqn. (2113)).

In the subcooled boiling model, it is assumed that the liquid is in full contact with the wall and is superheated first before vapor can be generated. However, if the wall begins to dry out, the factor, $K_{dry}$, can be specified to represent the wall contact area fraction for the vapor.

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:

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

(2114)

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:

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

(2115)

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, $s_{quench}$:

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

(2116)

The correction factor is defined as

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

(2117)

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

If other phases are present in the flow, they are assumed not to contact the wall and, therefore, have no influence on the wall boiling process.

Only the liquid phase l, and possibly the vapor phase g, are assumed to be in contact with the wall, as the wall contact area fraction $K_{dry}$ indicates.

The combined models for subcooled boiling with possible dryout (see Eqn. (2111)) fits into the multiphase energy balance at the wall (see Eqn. (2563)) as follows:

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

(2118)

where:

• ${{\dot{q}}^{"}}_{wl}$ is the heat flux between the wall and the liquid phase:

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

(2119)

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

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

(2120)

• ${{\dot{q}}^{"}}_{wg}$ is the heat flux between the wall and gas phase:

$${{\dot{q}}^{"}}_{wg}=K_{dry}{{\dot{q}}^{"}}_{dry}$$

(2121)

You can override any of the standard models for ${{\dot{q}}^{"}}_{wl}$, ${{\dot{q}}^{"}}_{w\lg }$, and ${{\dot{q}}^{"}}_{wg}$ with your own models.