This model describes phase change induced by interphase heat transfer; it may be used to simulate boiling and condensation, or melting and solidification. For example, it may be used to model condensation of saturated vapor bubbles in sub-cooled liquid, or evaporation of saturated bubbles in superheated liquid.
This section discusses the theory. For modeling information, see Thermal Phase Change Model.
It is essential to consider the heat transfer processes on each side of the phase interface. Hence, the Two Resistance model for interphase heat transfer must be used in conjunction with the Thermal Phase Change model. For details, see The Two Resistance Model.
Recall that, in this case, the sensible heat flux to phase 
from
the interface is:
 | (5–251) |
and the sensible heat flux to phase 
from the interface
is:
 | (5–252) |
and 
are the phase 
and phase 
heat transfer
coefficients respectively.The interfacial temperature is determined from considerations of thermodynamic equilibrium. Ignoring effects of surface tension on pressure, assume
, the saturation temperature.
Note that this is in contrast to the case where there is no
mass transfer. In that case, the interfacial temperature is determined
from the sensible heat balance 
.
In the case of interphase mass transfer, the interphase mass transfer is determined from the total heat balance, as follows.
Total heat flux to phase 
from the interface:
 | (5–253) |
Total heat flux to phase 
from the interface:
 | (5–254) |
denotes mass flux into phase 
from
phase 
.
and 
represent interfacial values of enthalpy carried
into and out of the phases due to phase change, see below for details.
The total heat balance 
now determines the interphase mass flux:
 | (5–255) |
The secondary heat flux term must be modified in order to take account of the discontinuity in static enthalpy due to latent heat between the two phases. This is achieved using a modification of the upwind formulation Equation 5–230, due to Prakash [53]. In this formulation, the bulk fluid enthalpy is carried out of the outgoing phase, as in the default upwind formulation. However, the saturation enthalpy is carried into the incoming phase. Thus:
 | (5–256) |
Wall boiling starts when the wall temperature becomes sufficiently large to initiate the activation of wall nucleation sites. This activation temperature is typically a few degrees above the saturation temperature. However, at this stage, the average temperature of the liquid in the vicinity of the heated wall is still well below the saturation temperature, hence in the sub-cooled boiling regime.
Evaporation starts in the microscopic cavities and crevices, which are always present on the solid surface. Liquid becomes supersaturated locally in these nucleation sites, leading to the growth of vapor bubbles. The bubbles detach when they are sufficiently large that external forces (inertial, gravitational, or turbulent) exceed the surface tension forces that keep them attached to the wall. As a bubble departs from the wall, it is displaced by subcooled liquid, after which the nucleation site is free to create another bubble. In regions of the wall not affected by bubble growth, the wall heat transfer to the liquid may be described by single phase convective heat transfer.
It is clear that detailed physics of bubble growth is very complex, and occurs on very small length scales in the vicinity of the wall. It is unrealistic to model the detailed physics in a phase-averaged Eulerian multiphase model. It is also unrealistic to resolve the small length scales with ultra-fine meshes. The model described here is a so-called mechanistic model, which aims to model the important physical sub-processes using engineering correlations. The model is a sub-grid scale model, in the sense that the complex physics is assumed to take place in a vicinity very close to the wall that is smaller than the mesh resolution at the wall.
The first and most well-known model of this kind was formulated by Kurul and Podowski (1991) [159], from the Rensselaer Polytechnic Institute. It is known as the RPI model. In this model, a number of the sub-models of the overall mechanistic model were taken from correlations originally developed for exploitation in one-dimensional thermo-hydraulic simulation methods. This model was implemented in CFX-4, with the near-wall distance taking the place of the centerline wall distance in the one-dimensional models. Unfortunately, this lead to results that were strongly mesh-dependent.
In the implementation in CFX-5.7.1, Egorov et al. formulated modifications of the one-dimensional correlations with the aim of restoring mesh independence to the results. This modification of the original RPI model is described here.
The following topics are discussed:
For more information about the wall boiling model and its usage, see Wall Boiling Model in the CFX-Solver Modeling Guide.
A fundamental feature of the mechanistic model of wall nucleation is the algorithm for deciding how the wall heat flux is to be partitioned among the separate physical processes of evaporation and sensible heating of the fluids in contact with the wall. In regions of the wall not influenced by nucleation sites, it is sufficient to consider the wall heat flux as contributing solely to single-phase liquid convective heat transfer. However, in the vicinity of the nucleation sites, some of the heat contributes to vapor production, and the remainder to super heating of the liquid phase as it displaces the rising bubbles. This latter process is known as quenching.
Hence, the mechanistic heat partitioning model has the following structure:
where 
is the heat
flux corresponding to convective heat transfer, 
is the evaporation heat flux, and 
is the heat flux corresponding to quenching.
The heat partitioning model considers the whole wall surface area as being separated into three fractions:
Fraction
is the fraction of wall area covered by a liquid film, influenced by vapor bubbles formed on the wall.Fraction
is the fraction of wall area that is in direct contact with vapor and therefore subject to convective heat transfer to vapor.Fraction
is the rest of the wall area, with 
.
For details on the calculation of these wall area fractions, see Area Influence Factors.
The wall area fraction 
represents the part of the wall surface that does
not 'feel' the presence of the vapor phase. The wall heat flux for
this part of the surface is modeled in a similar way as for the single-phase
flow of pure liquid, by using the turbulent wall function procedure
in the case of turbulent flow. This flux, called the convective
heat flux 
, can be correlated for turbulent flow as:
where:
and 
are the convective heat fluxes to the liquid and vapor, respectively.
is the temperature
of the solid wall.
is the temperature
of the liquid at the wall.
is the temperature
of the vapor at the wall.
is the turbulent
heat transfer coefficient for liquid, which depends on the velocity field and
on the near-wall grid cell size. Kurul and Podowski (1991) [159] modeled 
using a one-dimensional Stanton number correlation.
Egorov and Menter (2004) [163] modeled 
using the turbulent wall function.
is the turbulent
heat transfer coefficient for vapor.
The wall area fraction 
represents the part of the wall surface that
exchanges heat with both phases. The already mentioned heat
flux 
comes from
this part of the surface and is consumed for evaporation of the initially
sub-cooled liquid:
 | (5–257) |
where 
is the evaporation mass transfer
rate per unit wall area, and 
and 
are the specific enthalpies of the saturated vapor
and sub-cooled liquid, respectively.
The wall heat flux that passes to the liquid through
applies after a bubble departs and
before the next bubble forms at the same nucleation site. This additional
mechanism of heating the liquid phase is called quenching, and is
modeled as:
The area fraction values 
and 
play an important role in the heat-partitioning
model. They are related to the nucleation site density per unit wall
area 
and to the influence area of a
single bubble forming at the wall nucleation site. The latter value
is modeled by introducing the bubble departure diameter value 
, which can generally serve as a length scale of
the wall boiling mechanism. The RPI model assumes that the diameter
of the bubble influence zone is twice as large as 
, therefore the non-dimensional area fraction of
the bubble influence is:
 | (5–258) |
with 
used as shown in Equation 5–262.
Here, the diameter of the bubble influence is implicitly twice the bubble departure diameter. Note that the latter can be controlled via the Bubble Departure Diameter settings (see RPI Model in the CFX-Solver Modeling Guide).
It should be noted that, for the evaporation rate 
and corresponding
heat flux 
(Equation 5–257), the upper limit of Equation 5–258 for bubble influence area
fraction does not apply. Instead of using a factor of 
as defined by Equation 5–258, the 
value is directly modeled
as being proportional to the nucleation site density 
.
As mentioned in the previous section, the two most important
parameters governing the heat partitioning model are the nucleation
site density 
and the bubble departure diameter 
. In the RPI wall-boiling model they are correlated
to the wall superheat 
and to the
near-wall liquid sub-cooling 
, respectively.
The following topics are discussed:
The model for wall nucleation site density adopted in the RPI model is that of Lemmert and Chawla (1977) [164]:
Note that the wall superheat in the above equation cannot be negative. Its negative value means that the wall temperature drops below the saturation temperature, where there is no boiling and the heat partitioning model is not used.
This model was implemented in CFX-4 with parameter 
, giving an overall
correction factor of 
. Egorov and Menter (2004) [163], conjecturing
that this was a deliberate alteration by the RPI group, related to
the corresponding factor in the bubble waiting time model. Egorov
and Menter also reformulated the correlation as follows:
where 
and 
. This formulation
avoids fractional powers of physical dimensions, hence is more amenable
to the use of different unit systems.
Alternative correlations have been proposed for nucleation site density by Kocamustafaogullari and Ishii (1983) [160].
Kurul and Podowski (1991) [159] adopted the correlation for bubble departure diameter due to Tolubinski and Kostanchuk (1970) [165]:
 | (5–259) |
The parameters of the model are dimensional (
, 
, 
) and are chosen
to fit pressurized water data. Hence the model is clearly not universal.
Negative liquid sub-cooling is possible here, where it means onset
of the bulk boiling. Limiting the bubble departure diameter applies
to this situation and prevents 
from growing
too high.
Note that the model is strongly dependent on a liquid temperature
scale. In the original experimental data, this was taken as the pipe
center-line temperature. CFX-4 used the cell-center value in near
wall cells, but this proved to give mesh-dependent results. Egorov
and Menter restored mesh independence by using the logarithmic form
of the wall function to estimate the liquid temperature, 
, at a fixed value of 
. Ansys CFX uses this method, with a user-modifiable
value of 
.
The computation of the evaporation rate 
requires an additional
model parameter, namely the frequency of the bubble detachment from
the nucleation site. The model adopted by Kurul and Podowski is that
due to Cole (1960) [166]:
Note that, due to its dependence on gravity, this correlation
is taken from pool boiling. It is simply estimated as the bubble rise
velocity divided by the bubble departure diameter. The drag coefficient
factor 
was taken
to be unity by Ceumern-Lindenstjerna (1977) [169].
Kurul and Podowski (1991) [159] employed the model of Tolubinski and Kostanchuk (1970) [165]. This fixes the waiting time between departures of consecutive bubbles at 80% of the bubble detachment period:
 | (5–260) |
The numerator in this equation is adjustable in Ansys CFX.
Recall from Partitioning of the Wall Heat Flux that Kurul and Podowski
assumed a diameter of influence of a nucleating bubble equal to twice
the bubble departure diameter 
. Encoding
this as a user-modifiable parameter 
(default value = 2), the area fraction
of the bubble influence is given by:
with 
used as shown in Equation 5–262.
The area fraction 
is computed as follows:
 | (5–261) |
where:
is the vapor volume fraction.
is the critical volume fraction above which convective heat transfer to vapor is important.
is the fraction of wall area that is in direct contact with vapor and therefore subject to convective heat transfer to vapor.
The area fraction 
, which is subject to single phase liquid convective heat
transfer, is limited from below by a small value, so its actual form
is:
As discussed in Partitioning of the Wall Heat Flux, single-phase convective heat transfer to the liquid phase is modeled using the turbulent wall function (Egorov and Menter 2004 [163]).
This replaces the mesh dependent Stanton Number correlation originally employed by Kurul and Podowski (1991) [159].
The single-phase convective heat transfer to the vapor phase is:
As discussed in Partitioning of the Wall Heat Flux, quenching heat transfer to the liquid phase in the area of influence of the vapor phase is modeled using a quenching heat transfer coefficient:
In order to close the model, the quenching heat transfer coefficient 
, participating in the quenching heat flux to liquid,
must be defined. This value depends on the waiting time between the
bubble departure and the next bubble formation.
With this value, the quenching heat transfer coefficient is correlated as:
where 
is the liquid temperature conductivity coefficient. (Mikic and Rohsenow
1969 [167],
Del Valle and Kenning 1985 [168]).
As for the case of bubble departure diameter, Egorov and Menter
(2004) [163]
used the logarithmic form of the wall function to estimate the liquid
temperature 
at a fixed 
value of 250.
Knowing the bubble departure frequency, as well as the bubble size and the nucleation site density, one can obtain the evaporation rate as a product of the bubble mass, the detachment frequency and the site density:
This was the form adopted by Kurul and Podowski (1991) [159].
Egorov and Menter expressed the evaporation rate in terms of the
non-limited area fraction 
:
(unlike 
, 
can exceed 1). In this case, the evaporation
rate obtains the form:
which is an extension of the Kurul and Podowski expression and coincides with it when the Parameter
is equal to 2 (default value).
In the final form of the evaporation rate, the area fraction
factor 
is limited by:
 | (5–262) |
defaults to 0.5.
The wall heat flux partition discussed in Partitioning of the Wall Heat Flux, together with the choice of
submodels described in Sub-models for the Wall Boiling Model, gives the total
heat flux as a complex nonlinear function of the wall superheat 
In the case of an isothermal wall, the four components of the
heat flux are computed explicitly as functions of the given wall superheat.
However, in the case of a wall with specified total heat flux, or
specified wall heat transfer coefficient and outside temperature,
the above equation determines 
implicitly as a function of the total heat
supplied to the wall.