17.5.16. Wall Boiling Models
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 17–119), momentum
(Equation 17–120), and energy (Equation 17–126). The wall boiling
phenomenon is modeled by the RPI nucleate boiling model of Kurual and Podowski  and an extended formulation for the departed nucleate boiling regime (DNB) by Lavieville et al .
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, please refer to Including the
220.127.116.11. RPI 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:
The heated wall surface is subdivided into area , which is covered by nucleating bubbles and a portion , which is covered by the fluid.
; The convective heat flux is expressed as
where is the single phase heat transfer coefficient, and and
are the wall and liquid temperatures, respectively.
; The quenching heat flux models the cyclic averaged transient
energy transfer related to liquid filling the wall vicinity after
bubble detachment, and is expressed as
Where is the conductivity, is the periodic time, and is the diffusivity.
; The evaporative flux is given by
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:
; Area of Influence
Its definition is based on the departure diameter and the nucleate site density:
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:
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 :
and is the subcooled Jacob number defined as
; Frequency of Bubble Departure
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) 
; Nucleate Site Density
The nucleate site density is usually represented by a correlation based on the wall superheat. The general expression is of the form
Here the empirical parameters from Lemmert and Chawla  are used,
where and . Other formulations are also available,
such as Kocamustafaogullari and Ishii  where
Where is the bubble departure diameter and the density function is defined as
; Bubble Departure Diameter
The default bubble departure diameter (mm) for the RPI model is based on empirical correlations  and is calculated as
while Kocamustafaogullari and Ishii  use
with being the contact angle in degrees.
18.104.22.168. Non-equilibrium Subcooled Boiling
When using the basic RPI model, the temperature of the vapor is not calculated, instead it is fixed at the saturation temperature. In order to model different boiling regimes like DNB and critical heat flux, it is necessary to include the vapor temperature in the solution process. The wall heat partition is now modified as follows:
Here is the diffusive heat flux of the vapor bubble phase,
is the vapor heat transfer coefficient based on turbulent wall functions. The function depends on the local liquid volume fraction with similar limiting values as the liquid volume fraction. Lavieville et al  proposed the following expression:
Here, the critical value for the vapor fraction is
22.214.171.124. Interfacial Momentum Transfer
The interfacial momentum transfer may include four parts: drag, lift,
mass and turbulent drift forces (all described in Conservation virtual
Equations, Interphase Exchange Coefficients, and Turbulence Models. In
the wall boiling models, the virtual mass force is modeled using the standard correlation implemented in the Eulerian multiphase model within ANSYS FLUENT, while specific sub-models have been implemented for drag, lift, and turbulent drift forces. Also, user-defined options are available for both drag and lift forces.
126.96.36.199.1. Interfacial Area
The interfacial area is an important parameter for the drag and the heat transfer process. For dispersed boiling, the interfacial area, based on the diameter of the bubble, would be enough. However, as bubble coalescence takes place, this needs to be modified. The following options are included:
188.8.131.52.2. Interfacial Drag Force
The interfacial drag force is calculated using the standard model described in Interphase Exchange Coefficients (and defined in the context
of the interfacial area in Equation 17–290) is of the general form
Where the drag coefficient is determined by choosing the minimum of the viscous regime and the distorted regime , defined as follows:
The bubble diameter can be a constant value, a UDF, or a correlation function of local subcooling :
184.108.40.206.3. Interfacial Lift Force
The coefficient for the interfacial lift force is calculated using the correlation proposed by Moraga et al. :
Where . The lift coefficient combines the opposing action of two lift forces: the classical aerodynamics lift force resulting from interaction between bubble and liquid shear, and the lateral force resulting from interaction between bubble and vortexes shed by bubble wakes. Here is the bubble Reynolds number, and
is the bubble shear Reynolds number.
The formulation proposed by Tomiyama et al.  is also available with
the lift coefficient expressed as , where
where is the bubble Reynolds number and
is the Eötvos number, with as the gravitational acceleration and
the surface tension number.
220.127.116.11.4. Turbulence Drift Force
In the ANSYS FLUENT Eulerian multiphase model, the general correlation for turbulence drift force (turbulent dispersion) is based on Simonin . Due to numerical instabilities this force is now included in the Rhie & Chow interpolation  for the volume flux calculations.
Simonin’s  approach can be also used for the boiling model. However, for completeness of the RPI model, the default for the turbulent drift force is given by
Where the turbulent dispersion coefficient is, by default, set to 1.0
18.104.22.168. Interfacial Heat Transfer
22.214.171.124.1. Vapor to Liquid Heat Transfer
As the bubbles depart from the wall and move towards the subcooled region, there is heat transfer from the bubble to the liquid, that is defined as
Where is the interfacial area defined by Equation 17–291 and
is the heat transfer coefficient based on the Ranz-Marshall correlation 
126.96.36.199.2. Superheated Liquid to Vapor Heat Transfer The interface to vapor heat transfer is calculated using the constant time
method . It is assumed that the vapor scale return to saturation
retains the saturation temperature by rapid evaporation/condensation. The formulation is as follows:
Where is the time scale set to a default value of 0.05 and is the isobaric heat capacity.
188.8.131.52. Mass Transfer
184.108.40.206.1. Mass Transfer From the Wall to Vapor The evaporation mass flow is applied at the cell near the wall and it is
Equation 17–303 derived from the evaporation heat flux,
220.127.116.11.2. Interfacial Mass Transfer
The interfacial mass transfer depends directly on the interfacial heat transfer. Assuming that all the heat transferred to the interface is used in mass transfer (i.e. evaporation or condensation), the interfacial mass transfer rate can be written as: