We investigate the present capabilities of CFD for wall boiling. The computational model used combines the Euler/Euler two-phase flow description with heat flux partitioning. Very similar modeling was previously applied to boiling water under high pressure conditions relevant to nuclear power systems. Similar conditions in terms of the relevant nondimensional numbers have been realized in the DEBORA tests using dichlorodifluoromethane (R12) as the working fluid. This facilitated measurements of radial profiles for gas volume fraction, gas velocity, liquid temperature, and bubble size. Robust predictive capabilities of the modeling require that it is validated for a wide range of parameters. It is known that a careful calibration of correlations used in the wall boiling model is necessary to obtain agreement with the measured data. We here consider tests under a variety of conditions concerning liquid subcooling, flow rate, and heat flux. It is investigated to which extent a set of calibrated model parameters suffices to cover at least a certain parameter range.
Subcooled flow boiling occurs in many industrial applications where large heat transfer coefficients are required. However, the efficient heat transfer mechanism provided by vapor generation is limited at a point where liquid is expelled from the surface over a significant area. This occurs at the critical heat flux where the heat transfer coefficient begins to decrease with increasing temperature leading to an unstable situation. In this event a rapid heater temperature excursion occurs which potentially leads to heater melting and destruction. For a given working fluid, the critical heat flux depends on the flow parameters as well as the geometry of the flow domain. The verification of design improvements and their influence on the critical heat flux requires expensive experiments. Therefore, the supplementation or even the replacement of experiments by numerical analyses is of high interest in industrial applications.
In the past, many different empirical correlations for critical heat flux were developed and fitted to data obtained from experimental tests. These have been implemented mainly in purpose-specific 1D codes and applied for engineering design calculations. However, these correlations are valid only in the limited region of fluid properties, working conditions, and geometry corresponding to the tests to which they were fitted. Using large look-up tables based on a great number of experiments, a significant range of fluid properties and working conditions can be covered. But this method is still limited to only that specific geometry for which they were developed. Independence of the geometry can only be achieved by the application of CFD methods. Existing CFD models, however, are not yet able to describe critical heat flux reliably. A precondition would be the complete understanding and simulation of boiling as a preliminary state towards critical heat flux.
For engineering calculations, currently the most widely used CFD approach to model two-phase flows with significant volume fractions of both phases is the Eulerian two-fluid framework of interpenetrating continua (see, e.g., [
For the case of boiling flows, where heat is transferred into the fluid from a heated wall at such high rates that vapor is generated, additional source terms describing the physics of these processes at the heated wall have to be included. A CFD wall boiling model following the lines of Kurul and Podowski [
An application of continuing high interest is the thermal hydraulic flow in a nuclear reactor. However, typical flow conditions encountered in this application do not particularly lend themselves to experimental investigation. High pressure, high temperature, narrow channels, and small expected sizes of steam bubbles represent significant challenges for measurements. The use of refrigerants can greatly relieve this burden. In the French DEBORA tests [
The applicability of CFD models to the DEBORA tests was recently studied in Krepper and Rzehak [
In the present work, the previous methodology is applied to a number of further test cases for varied conditions of liquid subcooling, flow rate, and heat flux. In this way, the robustness of the model formulation can be assessed and the (in)dependence of model parameters on the experimental conditions be investigated.
The paper is organized as follows. A brief summary of the DEBORA test facility and selected data is given in Section
A detailed description of the DEBORA test facility can be found in Manon [
Sketch of the DEBORA test geometry.
Table
System parameters for the selected test cases,
Mass flow rate |
Wall heat flux |
Inlet subcooling |
Inlet temperature |
Outlet equilibrium vapor quality | |
---|---|---|---|---|---|
P26-G2-Q74-T16 | 15.7 | 70.7 | 0.0848 | ||
P26-G2-Q74-T18 | 17.9 | 68.5 | 0.058 | ||
P26-G2-Q74-T20 | 2000 | 74 | 19.8 | 66.6 | 0.0324 |
P26-G2-Q74-T26 | 25.7 | 60.8 | −0.0205 | ||
P26-G2-Q74-T28 | 27.6 | 58.3 | −0.0719 | ||
| |||||
P26-G3-Q118-T16 | 16.0 | 70.7 | 0.1056 | ||
P26-G3-Q118-T20 | 3000 | 118 | 20.2 | 66.2 | 0.0483 |
P26-G3-Q118-T24 | 24.2 | 62.2 | −0.0027 | ||
P26-G3-Q118-T28 | 27.6 | 58.3 | −0.0523 | ||
| |||||
P26-G3-Q129-T29 | 129 | 28.6 | 57.9 | −0.0334 | |
P26-G3-Q139-T28 | 3000 | 139 | 27.8 | 58.6 | 0.0055 |
P26-G3-Q148-T28 | 148 | 28.3 | 58.2 | 0.0259 |
To compare results obtained for the DEBORA tests to other typical experiments and applications, where the working medium is water at different pressure levels, the values of the relevant dimensionless groups have to be considered. For boiling phenomena, the pipe Reynolds number, the liquid-to-gas density ratio, the Jakob number, and the Boiling number play a role. Focusing on the bubble dynamics furthermore the bubble Reynolds number, the Eötvös number, and the Morton number are important. Conditions at which the DEBORA tests were performed have been chosen to match values of these dimensionless parameters to those for water at conditions typical for pressurized water reactors. The replacement of water by R12 allows measurements at more convenient pressures and temperatures. Also advantageous is the possible increase of pipe diameter which enables the measurement of radial profiles.
For R12 liquid and vapor, the relevant material properties were taken from the National Institute of Standards and Technology (NIST) Standard Reference Database on Thermophysical Properties of Fluid Systems (
Material properties of R12 and water at
Material | R12 |
---|---|
Pressure condition (MPa) | 2.62 |
Saturation temperature (K) | 360 |
Surface tension (N m−1) | 0.00180 |
Enthalpy of vaporization (J kg−1) | 293 × 103 |
Vapor density (kg m−3) | 0.172 × 103 |
Liquid density (kg m−3) | 1.02 × 103 |
Liquid specific heat capacity (J kg−1 K−1) | 1.42 × 103 |
Liquid viscosity (kg m−1 s−1) | 89.5 × 10−6 |
Liquid thermal conductivity (W m−1 K−1) | 0.0457 |
The general equations for diabatic two-phase flow in the Euler/Euler framework of interpenetrating continua have been reviewed in many places before (e.g., [
The first major block is the wall boiling model describing vapor generation at the wall and transfer of sensible heat to the liquid. Here, the CFX model closely follows the heat flux partitioning approach of Kurul and Podowski [
A second issue is the modeling of interfacial area that determines the exchange of mass, momentum, and energy between the phases. For the bubbly flow regime, it is convenient to use an equivalent Sauter diameter and work with the bubble size. Obviously, in boiling flows the bubble size may change due to both condensation/evaporation and bubble coalescence/breakup. The importance of taking into account the latter processes has been shown in Krepper et al. [
Turbulent fluctuations are modeled by a shear stress turbulence (SST) model according to Menter [
For momentum exchange between the phases, finally, lift and turbulent dispersion forces are included in the model in addition to the ubiquitous drag force. For adiabatic flows, there is in addition also a force that pushes a bubble translating in an otherwise quiescent liquid parallel to a wall in close proximity away from this wall. Different models for this so-called wall force were investigated, but the influence turned out to be small. Therefore, in the present study, this force was neglected. In general the applicability of a wall force to flow boiling should be investigated further.
In boiling, heat is transported from the hot wall to the fluid by several different mechanisms. On parts of the wall, where no bubbles reside, heat flows directly to the subcooled liquid in the same way as that in single-phase flow. On parts of the wall where bubbles grow, heat is consumed by the vapor generation which occurs at the so-called nucleation sites. Moreover, there is a liquid mixing mechanism due to the bubbles which leave the wall. As a consequence of the recirculation around the detaching bubble, a cold liquid from the bulk of the flow is brought into contact with the hot wall which leads to additional cooling. This mechanism, which is obviously not present in single-phase flows, is termed quenching.
Accordingly, the given external heat flux
The turbulent convection heat flux is calculated in the CFX model version (see Wintterle [
The evaporation heat flux
Some more readily usable results are possible for restricted conditions, for example, for water at atmospheric pressure. However, for the present application to R12, unfortunately even such more specialized results are not available. Yet more complex is the dependence on the contact angle, which varies significantly not only with the material combination of working fluid and heating surface but also depending strongly on ill-characterized factors like surface roughness and contaminants (e.g., Griffith and Wallis [
With respect to the other variables, it may be noted that for each individual test, the values of flow rate and heat flux are constant along the pipe. The available data on axial dependence of the wall superheat (cf., Section
Common practice in engineering simulations of flow boiling largely relies on an experimental investigation of the bubble size at detachment for water at atmospheric pressure by Tolubinsky and Kostanchuk [
The approach taken in the present investigation is to allow a different calibration for different values of flow rate and heat flux. Eventually it will turn out that within certain limits the same calibration can be used even for different conditions. Detailed values will be given in Section
Concerning the nucleation site density, most available correlations are expressed in the form of power laws depending on the wall superheat as
Sketch of bubble nucleus.
An important variable characterizing the thermodynamic stability of such nuclei is the local equilibrium temperature at the curved liquid vapor interface [
The diversity of specific expressions for these correlations is likely related to the geometry of corrugations on a given heating surface which depends strongly on the processes that were used to finish the surface. These processes are very diverse and in most boiling experiments not specifically controlled. Moreover, a characterization of the resulting surface topography is rarely available.
Parameter variations showed that the assumed nucleation site density has almost no influence on the calculated liquid temperature, a small influence on the calculated gas volume fraction, but a strong influence on the calculated wall superheat
In terms of bubble detachment diameter and nucleation site density given by (
Since
The bubble detachment frequency
The quenching heat transfer coefficient is calculated using the analytical solution for one-dimensional transient conduction, as suggested by Mikic and Rohsenow [
To describe polydispersed flows within a purely Eulerian approach, a number of different (multiple) bubble size groups
For each size group, the equation of mass conservation assumes the form
For the homogeneous MUSIG model only one momentum and energy equation for the total amount of vapor is considered as well as the conservation equations of the liquid of course. In these equations, the total gas volume fraction
The net mass source for size group
When condensation or evaporation occurs, the volume fraction in size group
Written as a source term for size group
In principle
A validation of the above procedure against experimental data has been given by Krepper et al. [
To include the generation of vapor bubbles at the wall, an additional source term,
Liquid-phase turbulence is modeled by a shear stress transport (SST) model [
The effects of the dispersed gas bubbles on the turbulence in the continuous liquid phase are modeled by introducing appropriate source terms in the
The bubble-induced source
The bubble-induced source
An equivalent source term for the
A wall function for boiling flow is obtained by considering that the presence of the bubbles on the wall forces the liquid into a similar flow pattern as that observed in single-phase turbulent flow with wall roughness (e.g., Ramstorfer et al. [
The hydrodynamic roughness
The constant of proportionality is not known from theoretical considerations at present, but a value of
As shown in [
For momentum exchange between the phases, the Ishii and Zuber [
The volumetric source of gaseous momentum due to drag exerted by the liquid is given by
A lift force due to interaction of the bubble with the shear field of the liquid was first introduced to two-fluid simulations by Zun [
The turbulent dispersion force is the result of the turbulent fluctuations of liquid velocity. Burns et al. [
As noted in Krepper and Rzehak [
The models described in the previous section present a rather general framework that can be specialized to the simulation of any subcooled flow boiling problem. The necessary specifications to simulate the DEBORA test cases will now be described. These comprise prescription of flow domain and boundary conditions and specification of grid and bubble classes as well as calibration of model parameters.
The simulations are performed by ANSYS CFX 13. For details of the numerical procedures, we refer to the user guide [
The tests were simulated in a quasi-2D cylindrical geometry, that is, a narrow cylindrical sector with symmetry boundary conditions imposed on the side faces. The validity of this simplification has been verified by grid resolution studies and by comparison to a 3D simulation representing a 60° sector of the pipe.
At the upstream end of the pipe, an unheated flow development zone has been added to obtain at the beginning of the heated section a fully developed turbulent flow that is independent of the detailed conditions imposed at the inlet. The required length of this flow development zone was determined by examining the development of velocity and temperature as well as turbulent kinetic energy and dissipation rate to ensure that these did not vary anymore in the axial direction before entering the heated section. At the outlet at the top, a pressure boundary condition was imposed.
On the heated walls, boundary conditions for mass and energy equations are provided by the heat flux partitioning discussed in Section
All of these two-phase flow simulations have been carried out on a quite coarse grid for which the center of the grid cells adjacent to the wall has a nondimensional coordinate of
The necessity to recalibrate the boiling model parameters to the working fluid and heating surface of the experiment was discussed in a previous study [
For the model of bubble detachment size in (
Parameters used for the bubble detachment model.
|
|
| |
---|---|---|---|
P26-G2-Q74-T16 | 359.5 | 0.1 | 0.247 |
P26-G2-Q74-T18 | 359.2 | 0.35 | 0.237 |
P26-G2-Q74-T20 | 359.0 | 0.62 | 0.228 |
P26-G2-Q74-T26 | 356.4 | 3.23 | 0.165 |
P26-G2-Q74-T28 | 351.6 | 8.03 | 0.154 |
| |||
P26-G3-Q118-T16 | 359.3 | 0.3 | 0.152 |
P26-G3-Q118-T20 | 358.5 | 1.1 | 0.152 |
P26-G3-Q118-T24 | 355.9 | 3.7 | 0.134 |
P26-G3-Q118-T28 | 352.5 | 7.1 | 0.124 |
| |||
P26-G3-Q129-T29 | 353.8 | 5.8 | 0.142 |
P26-G3-Q139-T28 | 355.7 | 3.9 | 0.163 |
P26-G3-Q148-T28 | 356.4 | 3.3 | 0.161 |
Adapted parameters for different test series.
Tests |
|
Equation ( |
Equation ( |
|
| ||
---|---|---|---|---|---|---|---|
|
|
|
|
||||
P26-G2-Q74-Txx | 28, 26, 20, 18, 16 | 0.24 | 45 |
|
10 | 0.03125 | 0.5 |
| |||||||
28, | 0.015 | 1.0 | |||||
P26-G3-Q118-Txx | 24, | 0.155 | 31 |
|
10 | 0.015 | 1.0 |
20, | 0.03125 | 1.0 | |||||
16 | 0.03125 | 0.5 | |||||
| |||||||
P26-G3-Q129-T28 | 0.0075 | 1.0 | |||||
Q139, | 28 | 0.155 | 31 |
|
10 | 0.0075 | 1.0 |
Q148 | 0.0 | 1.0 |
Bubble size at detachment according to (
P26-G2-Q74-T16 to T28
P26-G3-Q118-T16 to T28
P26-G3-Q129-T29 to Q148
As discussed in Section
Wall superheat depending on the reference nucleation site density
In the present work, bubble coalescence and breakup are described by the models proposed by Prince and Blanch [
Radial profiles of bubble size for different coefficients
As discussed in Section
In this section, a more detailed discussion is given for a few selected tests, namely, P26-G2-Q74-T16, P26-G3-Q118-T16, and P26-G3-Q148-T28. As summarized in Table
Comparison of measured and calculated values for P26-G2-Q74-T16:
Gas volume fraction
Gas and liquid velocities
Radial bubble size profile
Bubble size distribution at four points
Temperature
Cross-sectionally averaged values
Comparison of measured and calculated values for P26-G3-Q118-T16:
Gas volume fraction
Gas and liquid velocities
Radial bubble size profile
Bubble size distribution at four points
Temperature
Cross-sectionally averaged values
Comparison of measured and calculated values for P26-G3-Q148-T28:
Gas volume fraction
Gas and liquid velocities
Radial bubble size profile
Bubble size distribution at four points
Temperature
Cross-sectionally averaged values
For the first two test cases, experimental gas volume fractions exhibit s-shaped profiles with an inflexion point which is absent for the third test case. In the simulations, the gas volume fraction profiles for all cases have a slope that decreases monotonously from the wall towards the center of the pipe. Cross-sectionally averaged values agree with the experimental ones for the first two cases but deviate for the last one. There does not appear to be a simple explanation for these deviations at hand. Gas velocities (part b of the figures) are slightly overpredicted for all cases. Average bubble size profiles (part c of the figures) agree with the data to a varying degree which is not surprising due to the rather crude modeling approach for coalescence and break-up rates. The calculated bubble size distributions at several radial positions (part d of the figures) show a rather narrow distribution near the pipe wall which broadens towards the center. This is due to the present modeling where all bubbles are generated with the same value of
For all test cases, overall reasonable agreement between experiment and simulation is found. The relative deviations in all other aspects of the comparison are in the 20 to 30% range which is comparable to other state-of-the-art investigations in the field.
In this section, tests in two series with varying subcooling for otherwise fixed parameters are compared, namely, P26-G2-Q74-T16 to P26-G2-Q74-T28, P26-G3-Q118-T16 to P26-G3-Q118-T28, and P26-G3-Q129-T28 to P26-G3-Q148-T28. The resulting changes in the profiles for gas volume fraction, bubble size, and liquid temperature at the end of the heated length (
Gas volume fraction, bubble size, and liquid temperature profiles for test series P26-G2-Q74-Txx.
Gas volume fraction, bubble size, and liquid temperature profiles for test series P26-G3-Q118-Txx.
Gas volume fraction, bubble size, and liquid temperature profiles for test series P26-G3-Qxx-T28.
Thermal conditions in the liquid at the measurement location depend on all three parameters inlet subcooling, flow rate, and heat flux. The calculated temperature profiles compare quite well with the experimental ones for the case where the latter are available. As a reference the calculated profiles are shown also for the two cases where data are not provided.
For all tests, the bubble size increases with increasing the distance from the wall despite the fact that the bulk liquid is subcooled. Clearly this must be due to coalescence of the bubbles. This phenomenon could not be captured by the monodisperse model approach of Krepper and Rzehak [
Comparing the radial gas volume fraction profiles with decreasing subcooling, a broadening of the wall peaking profile can be observed for both test series. This effect is due to a lower condensation rate in the bulk liquid as the subcooling decreases. Again this behaviour is described qualitatively correct by the present modeling approach where bubbles of all sizes move with the same velocity. A change from wall to core peaking profiles as for the test series considered by Krepper et al. [
Boiling at a heated wall has been simulated by an Euler/Euler description of two-phase flow combined with a heat flux partitioning model describing the microscopic phenomena at the wall by empirical correlations adapted to experimental data. Such an approach was previously used and adjusted to boiling experiments with water at a pressure of several MPa. We here have investigated the applicability and necessary readjustments for similar tests using R12 at the DEBORA facility. At the same time, the bubble size distribution in the bulk was described by a population balance approach by coupling the wall boiling model with the MUSIG model.
A critical review of the detailed correlations used in previous work shows that some of the parameters used in these correlations have to be carefully recalibrated for the present applications. The DEBORA tests provide a large body of information that can be used to this end. Quantities with a strong influence on the amount of produced steam are the bubble size at detachment and the nucleation site density. The former can be taken straight forwardly from the measurements. On the latter, unfortunately no direct information is available; however, by matching the temperature of the heated wall, this gap can be closed. Previous work [
The measured gas bubble size profiles show an increase of the bubble size with increased distance from the heated wall. A monodispersed treatment is not able to capture this phenomenon, but including polydispersity by means of a MUSIG approach and suitable models especially for bubble coalescence this phenomenon can be described. In contrast to a previously investigated series of test cases [
A complete polydispersed description requires that processes of coalescence/breakup and condensation/evaporation must be modeled explicitly. For the latter, a suitable model is readily obtained from first principles with the aid of a heat transfer correlation like that of Ranz and Marshall [
Bubble coalescence and breakup are heavily influenced by two-phase flow turbulence. Unfortunately in the literature, only few measurements of turbulent characteristics of two phase flow can be found and even less when boiling occurs. Furthermore, models of bubble-induced turbulence working well for air/water flow may fail for steam/water flow at higher pressure or for refrigerants. In the present work, the selection of a specific model for bubble effects on the turbulence is confirmed mainly by plausibility of the final results. Hence, a more systematic investigation of approaches to modeling bubbly turbulence would be desirable.
Finally, looking carefully at the figures showing the gas volume fraction profiles in the near wall region, the calculated gas volume fraction is systematically too large. Reasons could be a missing force pushing the bubbles away from the wall or the neglect of swarm effects in the models of drag and lift forces even at gas volume fractions around 50%. Furthermore, the application of the simple heat transfer correlation of Ranz and Marshall [
Overall, our results confirm the great potential of the Euler/Euler two-phase flow and heat flux partitioning models for the simulation of subcooled flow boiling in industrial applications while at the same time highlighting the need for specific model improvements in order to achieve highly accurate quantitative predictions.
Interfacial area density
Bubble-induced turbulence coefficient [
Drag coefficient
Lift coefficient
Specific heat capacity at constant pressure
Turbulent dispersion coefficient
Virtual mass force coefficient
Wall force coefficient
Shear-induced turbulence coefficient (
Bulk bubble diameter (m)
Bubble diameter perpendicular to main motion (m)
Bubble detachment diameter (m)
Pipe diameter (m)
Eötvös’number
Bubble detachment frequency (Hz)
Drag force
Lift force
Turbulent dispersion force
Virtual mass force
Wall force
Acceleration of gravity
Heat transfer coefficient for single-phase convection
Heat transfer coefficient for bulk evaporation/condensation
Heat transfer coefficient for quenching
Specific enthalpy
Jakob’s number
Thermal conductivity
Turbulent kinetic energy
Length scale (m)
Massflux
Morton’snumber
Nucleation site density (
Pressure (Pa)
Prandtl number
Wall heat flux
Heat flux due to single-phase convection
Heat flux due to quenching
Heat flux due to evaporation
Radial coordinate (m)
Reynolds’ number
Hydrodynamic wall roughness (m)
Time (s)
Waiting time (s)
Temperature (K)
Saturation temperature (K)
Liquid subcooling (K)
Wall superheat (K)
Wall temperature (K)
Velocity (m
Friction velocity
Velocity scale (m
Volume (
Axial coordinate (m)
Distance to the wall (m)
Volume fraction
Viscous length scale (m)
Turbulent dissipation rate
Temperature scale (K)
Dynamic viscosity
Kinematic viscosity
Density
Surface tension
Wall shear stress
Bulk
Due to single-phase convection
Due to evaporation
Gas
Interface
Liquid
Due to quenching
Saturation
Total
Wall
Nondimensional.
This work is funded by the Federal Ministry of Education and Research (Contract no. 02NUK010A).