The term "subcooled boiling" is used to describe the physical situation where the wall temperature is high enough to cause boiling to occur at the wall even though the bulk volume averaged liquid temperature is less than the saturation value. In such cases, the energy is transferred directly from the wall to the liquid. Part of this energy will cause the temperature of the liquid to increase and part will generate vapor. Interphase heat transfer will also cause the average liquid temperature to increase, however, the saturated vapor will condense. Additionally, some of the energy may be transferred directly from the wall to the vapor. These basic mechanisms are the foundations of the so called Rensselaer Polytechnic Institute (RPI) models.

In Ansys Fluent, the wall boiling models are developed in the context of the Eulerian multiphase model. The multiphase flows are governed by the conservation equations for phase continuity (Equation 14–193), momentum (Equation 14–194), and energy (Equation 14–197). The wall boiling phenomenon is modeled by the RPI nucleate boiling model of Kurual and Podowski [328] and an extended formulation for the departed nucleate boiling regime (DNB) by Lavieville et al [345].

The wall boiling models are compatible with three different wall boundaries: isothermal wall, specified heat flux, and specified heat transfer coefficient (coupled wall boundary).

Specific submodels have been considered to account for the interfacial transfers of momentum, mass, and heat, as well as turbulence models in boiling flows, as described below.

To learn how to set up the boiling model, refer to Including the Boiling Model.

According to the basic RPI model, the total heat flux from the wall to the liquid is partitioned into three components, namely the convective heat flux, the quenching heat flux, and the evaporative heat flux:

(14–506)

The heated wall surface is subdivided into area , which is covered by nucleating bubbles and a portion , which is covered by the fluid.

(14–507)

where is the single phase heat transfer coefficient, and and are the wall and liquid temperatures, respectively.

Where is the conductivity, is the periodic time, and is the diffusivity.

(14–509)

Where is the volume of the bubble based on the bubble departure diameter, is the active nucleate site density, is the vapor density, and is the latent heat of evaporation, and is the bubble departure frequency. These equations need closure for the following parameters:

Its definition is based on the departure diameter and the nucleate site density:

(14–510)

Note that in order to avoid numerical instabilities due to unbound empirical correlations for the nucleate site density, the area of influence has to be restricted. The area of influence is limited as follows:

(14–511)

The value of the empirical constant is usually set to 4, however it has been found that this value is not universal and may vary between 1.8 and 5. The following relation for this constant has also been implemented based on Del Valle and Kenning's findings [136]:

(14–512)

and is the subcooled Jacob number defined as:

(14–513)

where

Implementation of the RPI model normally uses the frequency of bubble departure as the one based on inertia controlled growth (not really applicable to subcooled boiling). [116]

(14–514)

The nucleate site density is usually represented by a correlation based on the wall superheat. The general expression is of the form

(14–515)

Here the empirical parameters from Lemmert and Chawla [354] are used, where and . Other formulations are also available, such as Kocamustafaogullari and Ishii [318] where

(14–516)

Here

Where is the bubble departure diameter and the density function is defined as

(14–517)

The default bubble departure diameter for the RPI model is based on empirical correlations [655] and is calculated in meters as

(14–518)

while Kocamustafaogullari and Ishii [317] use

(14–519)

with being the contact angle in degrees.

The bubble departure diameter in millimeters based on the Unal relationship [664] is calculated as

(14–520)

(14–521)

(14–522)

(14–523)

where is the flow pressure, is the wall superheat, is latent heat, is the near wall bulk velocity, and m/s. The subscripts , , and denote the solid material, liquid, and vapor phase, respectively.

When using the basic RPI model (RPI Model), the temperature of the vapor is not calculated, but instead is fixed at the saturation temperature. To model boiling departing from the nucleate boiling regime (DNB), or to model it up to the critical heat flux and post dry-out condition, it is necessary to include the vapor temperature in the solution process. The wall heat partition is now modified as follows:

(14–524)

Here , , and are the liquid-phase convective heat flux, quenching heat flux, and evaporation heat flux, respectively (described in detail in RPI Model). The extra heat fluxes are representing the convective heat flux of the vapor phase, and representing heat flux to any other possible gas phases in a system. These can be expressed as

(14–525)

(14–526)

Similar to the liquid phase , the convective heat transfer coefficients and are computed from the wall function formulations.

The function depends on the local liquid volume fraction with similar limiting values as the liquid volume fraction. Lavieville et al [345] proposed the following expression:

(14–527)

Here, the critical value for the liquid fraction is .

In wall boiling, the critical heat flux (CHF) condition is characterized by a sharp reduction of local heat transfer coefficients and the excursion of wall surface temperatures. It occurs when heated surfaces are no longer wetted by boiling liquid with the increase of vapor content. At critical heat flux conditions, vapor replaces the liquid and occupies the space adjacent to heated walls. The energy is therefore directly transferred from the wall to the vapor. In turn, it results in rapid reduction of the heat removal ability and sharp rise of the vapor temperature, and most importantly, the wall temperatures. In addition, wall boiling departs from the nucleating boiling regime, and the multiphase flow regime changes from a bubbly flow to a mist flow.

To model the critical heat flux conditions, the basic approach adopted in Ansys Fluent is to extend the RPI model from the nucleate boiling regime to critical heat flux and post dry-out conditions, while considering the following:

 

14.5.25.4.3. Flow Regime Transition

When wall boiling departs from the nucleate boiling regime and reaches the critical heat flux and post dry-out conditions, the multiphase flow regime changes from a bubbly flow to a mist flow. Consequently, the liquid phase switches from the continuous phase to the dispersed phase, while the vapor phase becomes the continuous phase from the originally dispersed phase in the bubbly flow regime. With the flow regime transition, the interfacial area, momentum transfer terms (drag, lift, turbulent dispersion, interfacial area, and so on), heat transfer and turbulence quantities will change accordingly.

To mimic the change of the flow regime and compute the interfacial transfers, the so-called flow regime maps, based on cross-section averaged flow parameters, are traditionally used in sub-channel one-dimensional thermal-hydraulic codes. In CFD solvers, the concept of flow regime maps has been expanded into a local, cell-based interfacial surface topology to evaluate the flow regime transitions from local flow parameters. The ensemble of all the computational cells with their usually simple local interfacial surface topologies can provide complex global topologies to represent the different flow regimes as the traditional sub-channel flow regime maps.

As a first step, this implementation adopts a simple local interfacial surface topology to control the transition smoothly from a continuous liquid bubbly flow to a continuous vapor droplet flow configuration [650], [267]. It assumes that inside a computational cell, the local interfacial surface topology contains multi-connected interfaces, and the flow regimes are determined by a single local flow quantity — the vapor volume fraction :

• Bubbly flow topology: the vapor phase is dispersed in the continuous liquid in the form of bubbles. Typically

• Mist flow topology: the liquid phase is dispersed in the continuous vapor in the form of droplets. Typically

• Churn flow: this is an intermediate topology between the bubbly and mist flow topology, where

The interfacial surface topologies are used to compute the interfacial area and interfacial transfers of momentum and heat. Introducing to represent interfacial quantities (interfacial area, drag, lift, turbulent drift force and heat transfer), then they are calculated using the following general form:

(14–530)

Here is computed using equation Equation 14–528 or Equation 14–529, but with different lower and upper limits of the breakpoints. Typically, the values of 0.3 and 0.7 are used and and are the interfacial quantities from bubbly flow and mist flow, respectively. They are calculated using the interfacial sub-models presented in Interfacial Momentum Transfer and The Heat Exchange Coefficient.

It may be noted that in the boiling models, the liquid is usually defined as the first phase, and the vapor as the second phase. Once this is defined, it remains unchanged with the flow regime transition. When and are calculated, however, the primary or secondary phases are switched. For , the liquid is treated as the primary phase, while the vapor is the secondary phase. Contrary to this, for , the vapor becomes the primary phase and the liquid is the secondary phase.

The interfacial momentum transfer may include five parts: drag, lift, wall lubrication, turbulent drift forces, and virtual mass (described in Interphase Exchange Coefficients). Various models are available for each of these effects, some of which are specifically formulated for boiling flows. Also, user-defined options are available.

 

14.5.25.5.2. Bubble and Droplet Diameters

 

14.5.25.5.2.1. Bubble Diameter

By default, Fluent uses the following model for bubble diameter as a function of the local subcooling, :

(14–531)

where:

As an alternative, the bubble diameter, , can be given by the Unal correlation [664]:

(14–532)

To use the Unal correlation, Equation 14–532 , you can use the following scheme command:

(rpsetvar 'mp/boiling/bubble-diameter-model 2)

To return to the default formulation, Equation 14–531 , you can use the scheme command:

(rpsetvar 'mp/boiling/bubble-diameter-model 1)

 

14.5.25.5.2.2. Droplet Diameter

When the flow regime transitions to mist flow, the droplet diameter can be assumed to be constant or estimated by the Kataoka-Ishii correlation [292] :

(14–533)

where:
	= vapor volumetric flux (superficial velocity)
	= local liquid Reynolds number
	= local vapor Reynolds number
	= liquid viscosity
	= vapor viscosity

To model boiling flows, turbulence interaction models are usually included in the multiphase turbulence models to describe additional bubble stirring and dissipation. As described in Turbulence Models , three options are available for boiling flows: Troshko-Hassan (default), Simonin et al, and Sato.