Introduction
Venturi nozzles are used widely in industrial environments nowadays, that can be easily found in industrial pipe systems, steam engines, jet engines, etc It is a fact that erosion is a quite common issue in the pipe systems and could be seen as one of the most long-term potential risks that cause serious damages to the system High-pressure gas, steam and oil leaks or broken pipe due to critical due to severe cracks are possible bad consequences that could be happened once the erosion occurs Some important factors related to the erosion include the material of pipe, water velocity and pressure, etc
When the water flows under high velocity and pressure working conditions, the erosion of pipe walls trends to accelerate generally due to the increased friction rate between dissolved solids in the water and the inner pipe walls Flashing flow is one of the most interested phenomena related the high-pressure factor
The “flashing flow” term means a phase change from liquid to vapor due to the depressurization related to the nonequilibrium phase change effects If the vaporization happens because local pressure drops below the vapor pressure of fluid, as well as, the and the downstream pressure keeps below the vapor
2 pressure of working material, the process condition is called “flashing”, and the outlet stream will be fully controlled by the dispersed phase [2].
Motivation
Flashing flow is one of the most important phenomena referred to multiphase mechanisms in industrial environments, usually found in piping systems
Due to the high empirical cost, the computational numerical solutions, specifically computational fluid dynamics (CFD) approaches are found to be a useful alternate method Generally, the thesis purpose is to discover the flashing flow behavior in entrance high pressure and temperature condition by using computational fluid dynamic (CFD) simulation method.
Objectives and scope of study
The thesis is focused on simulations of “flashing flow” phenomena in entrance high pressure and temperature condition based on the modified thermal phase change model proposed in Ref [1] There are 5 objectives covered in the thesis including:
• Evaluate the suitable turbulence model for the flashing flow phenomena
• Compare and discuss simulation results of pressure and void fraction to experiment
• Evaluate the bubble number density influence on pressure and void fraction
• Appraise the inlet turbulence intensity effect on simulation results
• Explain the physical behavior of the flashing flow phenomena by the global and local void fraction sections
Literature review
Experiment reviews
The flashing flow due to sudden depressurization and non-equilibrium phase change effects is one of the most important phenomena in industrial environments From the 1960s to the early 2000s, there were numerous experimental and numerical studies related to flashing flow phenomena
The flashing flow in a vertical circular convergent-divergent nozzle with a subcooled water entrance and 49 taps along the nozzle was investigated by Abuaf et al The experimental design concept was to measure net vapor generation rates under nonequilibrium conditions by using a steady water loop with the well-controlled flow and thermodynamic The simple geometry of the nozzle comprised approximately 0.0512 meters of the inlet and outlet diameter and 0.0256 meters of throat diameter With different conditions of the constant temperature of subcooled water entrance combined with pressure inlet and outlet, various test cases were performed and effects of various parameters were reported and very useful for any simulated approaches [3]
An experimental program using the Moby–Dick nozzle was conducted at the Centre d’Etude Nucleaire (CEA) de Grenoble, whose purpose is to study the flashing flow phenomena of two-phase critical flow in different conditions for the hypothetical Loss of Coolant Accidents (LOCA) study in Pressurized Light Water reactors The Moby-Dick nozzle consisted of a smooth convergent- divergent section with an approximate outlet angle of 7 degrees, and a relatively long cylindrical throat One of the popular nozzle dimensions included a total length of one meter with an inlet diameter of 66.7 millimeters and an outlet diameter of 73.7 millimeters Subcooled liquid materials with various subcooling degrees to saturation ones, as well as, with different vapor mass fractions were taken into account as the nozzle inlet conditions The continuous reduce of pressure downstream was found, in which a maximum flow rate was
4 obtained through the nozzle in the experiment The behavior of nozzle flow according to several results of critical mass flow rate, pressure, vapor fraction, etc was analyzed by different charts during the tests [4]
Investigation of a water and steam mixture through a naval nozzle was presented by Deich et al, in which a broad spectrum of gas and various outlet pressure conditions are applied The wet steam whose wetness ranged from 0.0 in a superheated vapor state to a maximum value of 0.83 was applied as the material for experiments The geometry of the nozzle consisted of a circular inlet section with a radius of 28 mm, connected to a cone having the length of 122 mm and an approximate angle of 3.3 degrees The fixed value of pressure mixture entrance at 1.2 bar was used, as well as, the outlet pressure was reduced down to various values from 0.1 bar to 0.95 bar As a result, the flashing flow occurred due to the high depressurization in the nozzle The static pressure along the nozzle, as well as, liquid and vapor mass flow rates were collected and be preferably considered as qualitative measures for the flow behavior [5]
Besides, the flashing flow was also considered to be one of the preferred topics in pipe blowdown experiments One of the famous pipe blowdown transients in the 1970s was presented by Edwards In the experiment, an initial subcooled liquid at a high pressure of about 7.0 MPa and a temperature of 502K corresponding to a subcooling of 56.8K was fulfilled in a long one-closed end pipe The outlet was controlled by a rupture disk allowing the rapid depressurization to the environment at atmospheric pressure The values of pipe diameter and length were 0.076 m and 4.096 m respectively The area of flow at the rupture was reduced by about 13 percent The pressure along the pipe and void fraction at different sections of the pipe were found These results were used to compare to estimated values in many research in the safety nucleation major [6]
An experiment called “Super-Canon” test program was performed at the Centre d’Etude Nucleaire (CEA) de Grenoble, which was similar to Edward’s pipe
5 blowdown experiment A horizontal pipe of 4.39 meters in length and an internal diameter of 0.1 m with a one-closed end was also used However, the outlet of the pipe was opened totally to the atmosphere environment An initial pressure of 15.0 MPa and a temperature of 507K which was equivalent to a subcooling of 42K were chosen as the material of the experiment The temperature, pressure, and void fraction at different positions of the pipe were found [7].
Numerical study reviews
Various numerical studies related to flashing scenarios were conducted with application to nuclear safety analysis In general, it could be divided into 4 types of numerical models for flashing flow including the homogeneous equilibrium model (HEM), non-homogeneous equilibrium model (NHEM), homogeneous non-equilibrium model (HNEM), and non-homogenous non-equilibrium model (NHNEM) approaches that are summarized in Table 1 below
Models and references Features Remarks
Equal temperature Acceptable agreement for low metability, low void fraction, and long pipes
Equal temperature Mechanical interaction is considered with a two-fluid model, but reliable closure models are needed
Modeling of thermal non- equilibrium by nucleation and interphase heat transfer models is recommended, and further model improvement and development need take more efforts
A two-fluid model is greater than a drift-flux model, but accurate modeling momentum interaction is a prerequisite
The two-phase flows are simplified to a pseudo or an equivalent single phase one by HEM models The equivalent one flows have average velocities and mean thermodynamic properties, which are derived by interpolation process between the saturated liquid and vapor ones, in which the equilibrium quality equation is applied
A generalized correlation for a simple homogeneous-equilibrium choked flow was presented by Lueng While experiencing an isentropic expansion with identical phasic velocities and temperature, the flashing two-phase mixture was considered as a single-phase compressible fluid Good agreements in normalized critical mass flux and critical pressure ratio for water and ten common fluids such as ammonia, nitrogen, propane at different stagnation pressure conditions were provided It was suitable for long pipes, where the time was enough for equilibration between the phases to happen [21] In contrast, for short pipe, where the time was not enough to reach the equilibration, there were differences between the empirical and numerical results The two-phase critical flow of steam-water mixtures in a short pipe had been investigated in various pressure conditions by Isbin et al The result showed that the obtained critical flows are greater than the preceding reported ones by others [10]
A study of the critical flow of steam-water mixture in blowdown pipe was performed by Ardron et al, which provided some limits of the HEM model in thermal equilibrium There was a good comparison with all available critical flow rate of steam-water mixtures evaluation in a restricted range of thermodynamic and geometry conditions However, some worse cases were found with low quality of flows, in which an influence of depressurization transient in the mixture was reported as one of the most important parameters that should be considered [22]
A steady one-dimensional separated two-phase flow model applied hydrodynamic as well as thermal non-equilibrium effects under quickly depressurizing were present was publicized by Richter In the first stage of flashing, when small bubbles’ dispersion was finely occurred in the liquid, in which hydrodynamic momentum equilibrium and thermal non-equilibrium assumptions were considered due to the limit of the interfacial area for heat transfer Whenever the bubbles grew and the void faction had a value more than 0.3, thermal equilibrium might be used for one of assumptions but the slip between the phases became critical and it would cause inaccuracies if the slip model between two phases was ignored [23]
1.4.2.2 Non-homogeneous equilibrium (NHEM) models
In the non-homogeneous equilibrium (NHEM) approach, the velocity of two phases became separated and interfacial slip or drift-flux models were applied to define the relative velocity between the two phases, however, the thermodynamic equilibrium condition was still assumed
The interfacial slip effect on bubbly two-phase flow through a sudden extension was researched by Attou et al The global mass, momentum, and energy conservation equations, as well as, the thermal equilibrium of the phases were considered Especially, the kinematic non-equilibrium effect had to be taken into account because of the different mechanical inertia of the phases To determine the role of mechanical non-equilibrium, two assumptions were considered An infinite momentum transfer coefficient and no momentum transfer between phases were assumed, in which ideal gas was applied With the first assumption, a higher-pressure recovery than experiments was found because of the faster liquid deceleration than in reality due to the lower gas inertia, which was opposite to the second assumption In the MFM assumption, the slower liquid deceleration leads to the lower pressure recovery than the experimental data [24]
To develop empirical or theoretical correlations for the slip ratio between two phases, there were several researched topics related to the NHEM approach An analytical model for predicting the highest flow rate of a two-phase mixture based on annular flow, steady linear velocities of each phase, and liquid-vapor equilibrium was conducted by Moody The result showed that the highest flow rate value was obtained when concerning the local slip ratio and static pressure for known stagnation conditions [13]
A two-phase critical discharge of a one-component mixture through a convergent nozzle model considering the interphase heat, mass, and momentum transfer rates was investigated by Henry et al The results showed a good agreement in critical flow rate with the data over broad stagnation conditions based on the comparison between the theoretical and experimental results in different materials including water, nitrogen, potassium and carbon dioxide [14] The mechanical interaction between the phases considering a separated flow model or two-fluid model should be applied in the flashing flow simulation [24]
1.4.2.3 Homogeneous non-equilibrium (HNEM) models
For the homogeneous non-equilibrium (HNEM) approach, a similar assumption to the homogeneous equilibrium model in velocity was taken into account However, the HNEM models are concerned with the non-equilibrium thermal effect on flows
A one-dimensional non-equilibrium relaxation model of flashing phenomena in liquid was presented by Downar-Zapolski et al According to them, the most critical character of flashing flows was thermal non-equilibrium due to nucleation delay and rate of vapor generation process as pressure drops, which greatly influenced the void fraction as well as the pressure and velocity distribution along a flow Besides, a correlation for the relaxation time which was a closure law for the homogeneous relaxation model (HRM) taken into
9 account to the non-equilibrium evaporation leading to the metastable liquid conditions was also provided The HRM prediction of the critical mass-flow rates and the pressure distributions had been validated by available experimental data [25]
The thermodynamic effects in cavitation bubble growth for hot water, liquid hydrogen, and nitrogen were proved by a numerical study of Kato et al It was shown that the model had to capture accurately the non-equilibrium nature of the flow by solving the combination of energy equation and the Raleigh’s equation [26]
A new model called Delay Equilibrium Model (DEM) was not yet widespread due to its complex assumption of three phases, in which the third phase was in thermal non-equilibrium with the saturated ones The purpose of the DEM model was to describe the flashing flow using the three-phase mixture including saturated liquid, saturated vapor, and metastable liquid A flashing model of a supersaturated liquid using the DEM approach had been developed by Lackme
It was shown that vaporization is not complete, which leads to approximate 90% of the liquid in a supersaturated state When vaporization was choked, the vaporized products were blown out at the local sonic velocity [27]
Computational fluid dynamics (CFD) models
In general, two important aspects must be considered in the CFD approach to model flashing flows including the phase change during the flashing process and the liquid-vapor interaction during the vaporization process Numerical
12 models can be divided into two main categories based on the change of phase during the flashing flow, comprising models accounting for the nucleation process and models neglecting the one and assuming constant bubble radius or number density, with a special quantity called artificial coefficients to control the nonequilibrium process Concerning the interaction between two phases during the vaporization process, the mixture approach, and the Eulerian multi- fluid approach, which is more general than the first one, are usually applied in the CFD approach The CFD works on flashing nozzle flow are also divided into two categories comprising thermal phase change models (TPCMs) and pressure phase change models (PPCMs) based on phase change models
In the thermal phase-change model, the main considered force factor for phase change is the temperature difference across the interface between two phases
In contrast, for the pressure phase change model approach, the pressure difference is applied instead of the previous one
In thermal phase change models, temperature is the main parameter used to define the fluid state If the temperature of the mixture is below the saturation temperature at a certain pressure value, the process will be considered an evaporation process, otherwise, a condensation
The transient flashing flow calculation of a three-dimensional CFD model using the commercial CFD-code CFX4.2 was presented by Maksic et al The simplified two-phase model considered two phase flow, as well as, the thermal non-equilibrium between the phases To verify the model ability, calculations of the flow through the BNL 237 nozzle [3] were carried out, in which the turbulence K-Ɛ model and the neglect of the influence of the dispersed vapor phase were applied In addition, the vapor generation and propagation were modelled by the bubble number density equations in the Jones’s model [35] and vapor mass conservation equations for the vapor phase The model results
13 showed a good agreement on mass flow rate with the experiment data Moreover, a pressure fluctuation profile and under-prediction of void fraction in the divergent section were also found [36]
The flashing flow process in the BNL nozzles [3] was modelled by Marsh and O’Mahony, in which the commercial CFD code, with a User-Defined Scalar (UDS) transport equation used to solve bubble nucleation and transport were used The two fluid model with separate mass, momentum and enthalpy conservation equations, inter-phase mass and momentum as well as energy transfer derived result from nucleation process and phase change process were considered However, the non-drag forces effect on the momentum exchange and heat transfer between phases were ignored The results show good agreements for almost cases except the Run 309 which ran under a high entrance pressure and a low outlet pressure [37] There was a main difference between the simulated results when comparing with simulation of Maksic’s research Specifically, the Blander and Katz model [38] used in Marsh and O’Mahony study was deduced from the classical nucleation theory for bulk nucleation and affected on both the walls and the entire remaining domain depending on local liquid superheat, which is more different than the nature of the Jones wall cavity model [35] applied in the preceding work
In pressure phase change models, pressure is the main parameter used to define the fluid state If the pressure of the mixture is below the saturation temperature at a certain pressure value, the process will be considered an evaporation process, otherwise, a condensation
The flashing flow phenomena simulation of steady flow through BNL nozzles [3] using the of Singhal et al’ cavitation model [39] available in Ansys Fluent 6.1 was performed by Palau-Salvador et al The authors found that pre-existing nuclei growth increased when the local pressure dropped below the saturation
14 pressure, and collapse in the reverse case The results show good agreements between simulation and experiment data in all cases With a difference less than 4% between the measured and calculated mass flow rates was achieved, the simple model was proposed to apply in steady flow case The authors also applied the model to the unsteady flow, and the simulation results obtained from the numerical model did not match with experimental data [40]
The two-phase critical flow simulation in the nozzles and breaks using another cavitation model called Schnerr and Sauer model [41] in the Ansys Fluent code, was presented by Ishigaki et al The Super Moby Dick (SMD) experiment [42] and the SGTR experiment [43] were used for the model validation The numerical results showed good agreements of the mass flow rates to the experimental results Besides, the authors also emphasized about the possibility of CFD code Ansys Fluent to simulate two-phase critical flows referred to the nuclear safety analysis [44]
A new model for flashing flow prediction using two-phase mixture with slip model approach was performed in Ref [1] In the simulation, the thermal phase change Lee model [45] was modify by combination of Clapeyron-Clausius equation and the formula of vaporization pressure [46] The Clapeyron-Clausius equation was used to convert the thermal phase model to pressure phase change model The model used pressure instead of thermal to define the flow states A new parameter called “accommodation” coefficient was taken into account for thermal non-equilibrium effects The model was validated against BNL experimental data [3] Simulated results showed good agreement of mass flow rate with the highest relative error below 6%, average void fraction and static pressure along nozzle to experiment [1]
A numerical study on heat transfer effects of cavitating and flashing flows was conducted by Jin et al The combination of a homogeneous mixture model in an
15 in-house CFD code, and the effects of both pressure and thermal difference on the interfacial mass transfer was applied in the model The results showed good agreement of pressure profiles and void fraction in two BNL nozzle cases including the Run 309 and the Run 268 [47]
Based on author’s knowledge, most of previous numerical solutions on flashing flow are focus on low Mach below 0.3, incompressible and mildly compressible flows However, the numerical researches for high compressible flow are still limited up to now The thesis aims to investigate the flashing flow behavior in high compressible flow based on homogeneous mixture model and validate with a Super Moby Dick nozzle experimental case [48] After considering the results of several cases applied thermal and pressure phase changes in the BNL nozzle test case 309 [3], which is described detailed in Chapter 3, the modified thermal phase change model provided in Ref [1] shows higher precise values in mass flow rate and average vapor fraction than the others Therefore, the model proposed in Ref [1] is applied in the thesis for simulating the flashing flow phenomena in high compressible flow.
Outline of the thesis
The content of the thesis consists of 5 chapters with the following structure: Chapter 1: Introduction and literature review
Chapter 2: Flashing flow phenomena and mathematical model
Chapter 3: Chosen phase change model
Chapter 4: Simulation of flashing phenomena in the Super Moby-Dick nozzle Chapter 5: Conclusion and further development
This chapter is focus on presenting the Super Moby-Dick experiment, providing the knowledge and mathematical model of flashing flow phenomena.
Super Moby-Dick experiment
Experiment Introduction
The Moby–Dick nozzle test program [4] was conducted at the Centre d’Etude Nucleaire (CEA) de Grenoble to find out the behaviors of two-phase critical flows under several test conditions The experiment results were recognized in the French Nuclear Thermal Hydraulic code CATHARE qualification [49] The two-phase critical flow studies were critical topics of analysis of hypothetical Loss of Coolant Accidents (LOCA) in Pressurized Light Water reactors
Figure 2.1: Super Moby-Dick experiment setup [48]
17 The primary loop is composed of a centrifugal pump with nominal point of
20 kg/s and 5 MPa, a preheater, a test section and a condenser The preheater has a maximum power of 3.5 MW and the 1m3 condenser can support a maximum pressure of 1.3 MPa The pressure at the inlet of the test section and the inlet temperature are regulated by a by-pass and two valves, and the electrical power of the preheater (0.1 o C maximum oscillation) [48].
Super Moby-Dick nozzle test section
Figure 2.2: The Super Moby-Dick nozzle geometry [48]
The Moby Dick nozzle components shown in Figure 2.2 above include a convergent section, a long cylindrical throat and a divergent section The divergent section has an approximate angle of aperture of 7 degrees
The nozzle operating condition is presented in the Table 2 below with the entrance temperature of liquid phase being approximate 9K lower than the saturation temperature based on inlet pressure
The nozzle geometry is presented in Figure 2.3 and its parameters are shown in the Table 3 below
Figure 2.3: Moby-Dick nozzle geometry [48]
Table 3: Moby-Dick geometry parameters
Experiment data
Three important parameters recorded in the paper include the mass flow rate of 10.3 kg/s, pressure, and average void fraction along nozzle
Pressure values were measured at several locations along the wall of the test section with transducers 1151 GP (0-20 MPa) The diameter of each pressure tap is 0.6 mm The transducers are calibrated with a high accuracy, therefore, the pressure chart presented in Figure 2.4 below is used as a critical parameter in the thesis
The method measured the void fraction along the nozzle is “Cross-sectional average void fraction” method The method formula is presented in Eq (1) below:
In which c s − ,A G , A L are the void fraction ratio, the area of the cross-section of the channel filled by the vapor phase and liquid phase, respectively
Figure 2.5: Cross-sectional average void fraction
The experimental void fraction chart is presented in the Figure 2.6 below:
Figure 2.6: Measured void fraction chart [48]
The process measuring void fraction is very difficult to evaluate and the averaging process in the cross section certainly led to a loss of accuracy, which is reported in the experiment paper From simulation results including periodical trend lines of residuals, and mass flow rates of inlet and outlet in section 4.2.1 below, the obtained simulated flow behaviors are not constant In addition, a remarkable disadvantage of the “Cross-sectional average void fraction” method is that the method should not be applied in the inconstant fluid behavior, which usually makes the void fraction measurement is incorrect Therefore, the
21 average void fraction will not be used as a critical parameter of validation in the thesis.
Flashing flow phenomena
The thermodynamic diagram of water phase change in the boiling process is shown in Figure 2.7 below
Figure 2.7: Flashing flow behavior: (a) thermodynamic diagram of phase change of water with equilibrium assumption and (b) start of vaporization [1]
At a constant pressure, the phase change process between the liquid and vapor phases occurs in a saturation region under thermal equilibrium conditions However, the non-equilibrium effects are important factors affected to the phase change of high-speed flows inside nozzles, for instance, ejector-based systems Therefore, it is impossible to use this diagram above for fluid dynamics prediction in the phase change region Specifically, sudden depressurization occurs when a subcooled water flow enters the nozzle, which leads to superheated water, and the phase change appears The rapid depressurization leads the liquids to be in metastable state until the phase change happens [50] Consequently, the nonequilibrium effect will critically affect to the bubble nucleation process [51] In general, the phase change mechanism driven by an abrupt pressure drop is divided into in three stages including formation of nucleation droplets, bubble growth, and the bubble boiling explosion The flashing phenomena of two-phase flows is usually found in industrial energy systems such as trilateral cycle power systems, supercritical CO2 power systems and ejector heat pumps [52]
In the first stage, small nuclei appearing in non-wettable rough wall cavities and in the bulk the formation of start growing after the pressure decreases below saturation pressure [1] Another explanation is that the nucleation takes place due to the difference temperature between the liquid phase and vaporization state, specifically, the saturated temperature at a given ambient pressure is significantly lower than the temperature of liquid phase In this stage, the bubble nucleation is critically affected by the superheated degree of the droplets [51] The surface tension forces play dominated factors that directly influence These forces limit the nuclei growth rate, therefore, this stage is also called as delay period [53] In the initiating nucleation process, the liquid superheat degree required for the nucleation is dependent on the depressurization rate, surface conditions and the flow characteristics The liquid superheat degree at the begin of flashing inception is noticeably smaller than the kinetic limit of nucleation in
23 slow phase transitions Typically, the initial bubble formation in a pipe is dominated by heterogeneous nucleation [54]
Once the bubble nuclei diameter crosses over the critical size, the following process will perform as the second stage of flashing The critical diameter of bubble in nucleation state can be estimated from the force balanced in which the spherical symmetric geometry is applied in bubble shape Two sub stages are considered in this stage are considered in this stage including the remain of bubble radius at the beginning of the growth stage, in which the bubble radius is still small and the surface tension is still dominated After the surface tension effect weakens and the thermal diffusion begins to dominate and control the bubble growth process, the bubble radius increasement will become significant The bubble grow rate in this stage increases significantly under high temperature working conditions [51] The pressure difference between the bubble surface and the surrounding water is one of the crucial keywords to describe the mixture flow properties In addition, the bubble growth rate in this state can be predicted in Ref [55]
The heat transfer around the bubble interface is the most important factor which dominate and control the final stage The heat transfer mechanism at water- bubble interface is critically influenced by the turbulence fluctuations and relative motion between the dispersed phase and the continuous phase [56] The bubble boiling explosion is broadly known as the main dominant mechanism for the fuel atomization after flashing happens [51] The flashing increases the liquid jets atomization performance, and the higher-quality liquid fuel atomization is an important keyword to acquire the better combustion efficiencies and lower the pollutant emissions [57].
Mathematical Model
Mass conservation
Where ,v , t , r are density (kg/m 3 ) and turbulent viscosity (m 2 /s), time coordinate (s) and radial coordinate (rad), respectively The subscripts symbols ofx, r ,m are axial component, radial component, and mixture.
Axial momentum conservation equation
In which ,v, are dynamic viscosity (Paãs), velocity vector (ms -1 ) and vapor volume fraction, respectively The subscripts symbols of dr_x,dr r_ ,k are axial component of drift velocity, radial component of drift velocity, and kth phase.
Radial momentum conservation equation
In which pis pressure (Pa).
Energy conservation
In which E, T ,nare internal energy (kg/m 2 s -1 ), temperature (K) and number of phases, respectively
In the governing equations above, the averaged-mass mixture velocity v m
(ms -1 ), the mixture density m (kg/m 3 ), the mixture dynamic viscosity m (m 2 /s), and the effective conductivity eff (Wm -1 K -1 ) are calculated as following formulas:
Where t is turbulence thermal conductivity (Wm -1 K -1 )
In the thesis, the work contribution due to shear stresses is neglected as the assumption applied in state-off-the-art mixture model.
Slip model
The slip model is formulated in term of the drift velocity,v dr p , The drift velocity between secondary phase velocity,v p and mixture velocity,v m :
The relative velocity between the primary phase ( ) q and secondary ( ) p , pq = p − q v v v , is used to drive Eq.(12) from Eq.(11):
The c k = k k / m stands for the mass fraction off phase k th
The Ansys Fluent software applies a modified version of Ref [58] to define v pq
, including a diffusion term for modeling the turbulence dispersion of dispersed phase
Where a , v m and D are acceleration vector (ms -2 ), the mixture turbulent viscosity (m 2 /s) and the Prandtl dispersion coefficients (m), respectively Particle relaxation time of secondary phase, p (s), is defined as follows:
Where p and d p are density (kg/m 3 ) and bubble diameter (m) of secondary phase, respectively q is defined as dynamic viscosity of primary phase (m 2 /s)
In Eq.(13), f drag is the drag function calculated in Ref [59]:
1 0.15 Re Re 1000 0.0183Re Re 1000 f drag +
Where Re is the relative Reynolds number function of the particle diameterd p
The Ansys Fluent software only takes care of the slip effect between the two phases and ignores other non-drag forces in the mixture model.
Phase change model
From the vapor transport equation:
The Clapeyron-Clausius equation is applied to adjust the Lee model [45], in which the original thermal phase change model is transformed to pressure phase change model
The change of phase source terms of Lee model [45] is presented below:
Where T * is the vapor temperature at the interface (K) and is assumed to be close to saturation temperature T sat (K) m ,R, M ,F ,Land A i are mass flow rate (kg/s), gas constant R (KJ/kmolK), molar mass (kg/kmol), the evaporation- condensation flux at flat interface (kg/m -2 s -1 ), latent heat (J/kg) and interfacial area density (m -1 ), respectively is an accommodation” coefficient and should be considered carefully dependent on operating conditions The subscript symbols of l, vare liquid and vapor phase
The Clapeyron-Clausius equation as in Eq.(19) shows the exchange between pressure and temperature at the saturation condition
From Eq.(19), the variation of pressure near saturation conditions can be derived as in the following equation:
Where P * (Pa) is the partial pressure at interface on the vapor side with value close to saturation pressure, P sat (Pa)
Finally, the mass source term for the vapor transport at the interface is defined as follows:
The interfacial area density is proposed by Liao and Lucas [60]:
In which , N b are pi constant and bubble number density The subscript symbol b means bubble
The formula of the vaporization pressure in Ref [46] is used to deal with the local turbulence effect in flashing flows:
In which k is turbulence kinetic energy (m 2 s -2 )
Finally, the mass source term becomes:
The sufficient small value of dp=P sat −P * is required in this model
This chapter is focus on choosing the phase change model applied in the master’s thesis The BNL nozzle experiment data are used as materials of the comparison process, in which several numerical approaches are performed.
BNL nozzle experiment
The experimental data of the BNL nozzle (Case 309) [3] is used to compare several numerical models that are unique or combination of thermal phase change model and pressure phase change model From that, the optimum phase change model solution is chosen in the thesis The BNL geometry presented below is measured in 2D Cartesian coordinate
Figure 3.1: Vertical circular convergent-divergent nozzle geometry [3] Table 4: BNL working condition
The mixture model with slip model is used The boundary conditions, setup condition in Fluent as well as numerical solutions are listed in Table 5, Table 6, and Table 7 below:
Table 6: BNL Ansys Fluent method
Velocity-Pressure Scheme PISO (Pressure-implicit with splitting of operators)
Transient formulation Second-order implicit
Approach Phase Change Models Reference
1 Lee thermal phase change model (UDF) [1]
2 Lee thermal phase change model (fluent code)
+ Schnerr and Sauer cavitation model (fluent code)
3 Lee thermal phase change model (UDF) +
Zwart-Gerber-Belamri cavitation model (fluent code)
4 Lee thermal phase change model (UDF) +
Schnerr and Sauer cavitation model (fluent code)
The Liao et Lucas results [60] and four numerical approaches are used to compare and choose the best solution up to now according to author The modified Lee thermal phase change model (UDF) was in Ref [1] The knowledge of the phase change model is presented in section 2.3.6 below.
Simulation results
Mass flow rate
Table 8: Mass flow rate comparison
Case Mass flow rate (kg/s) Relative error %
Pressure along nozzle
Figure 3.2: Absolute pressure at nozzle axis comparison.
Average vapor fraction along nozzle
Figure 3.3: Average vapor fraction comparison
Discussion
All numerical approaches were convergent, as well as, from comparing parameters including mass flow rate, pressure and average vapor fraction along nozzle of each method with experimental data, the model suggested in Ref [1] showed better results in mass flow rate (with relative error reached 0.2%) and in average vapor volume fraction at most of empirical points’ value than the others
From this point, the thesis will apply the model suggested in Ref [1] to the flashing flow simulation in the high pressure and temperature conditions The Supper Moby-Dick nozzle geometry and data will be used as materials for the thesis
Simulation setup
Meshing process
The first step is to create the geometry according to the experiment geometry data The software DesignModeler (in Ansys Workbench) is used to create the geometry for the simulation The Super Moby-Dick nozzle geometry is separated to 6 sections and dimensions of each section is listed in Table 9 below The simulated geometry dimension in millimeter is listed as following:
Figure 4.1: Nozzle geometry separated sections Table 9: Nozzle section dimension
Because the geometry is axisymmetric, just the half of simulation is simulated to save resources and reduce simulation time The export meshing shown in
35 Figure 4.2 below is used in the software Ansys Fluent 2020R2 to perform the simulation
In the thesis, four meshes with the mesh ratio of 1.5 are taken into account before deciding the most optimum mesh to solve the flashing flow problem The number of elements of each mesh is listed in the Table 10 blow
Mesh Number of elements Mesh ratio
All meshes that are chosen after selecting the final Y+ value in section 4.2.2 are validated by three important critical parameters including the skewness, the orthogonal quality and the aspect ratio The skewness and orthogonal quality of all mesh is in good range of spectrum of mesh metrics spectrum recommendation by Ansys To reduce simulated time consumption, the aspect ratio is allowed to be higher than the recommendation without effect the final results The maximum aspect ratio is placed on the cells near the wall, but in general, the average aspect ratio values of meshes are various from 10 to 17 The detailed information of meshing quality is presented in Table 11 below
Mesh Skewness Orthogonal quality Aspect ratio Min Max AVG Min Max AVG Min Max AVG
Material properties
Physical properties of materials including water and vapor data is taken from the software NIST The mini-REFPROP or “Reference Fluid Thermodynamic and Transport Properties Database” program (version 10.0) is one of useful tools to export thermodynamics properties of materials The software supports numerous types of pure fluids such as water, CO2, R134a, nitrogen, oxygen, methane, etc [62]
Figure 4.4: Water and vapor properties at saturation point
Mixture model
In the thesis, the mixture model in transient mode with slip velocity is used for the simulation Water and vapor are primary and secondary phases, respectively The UDF code proposed in Ref [1] is applied as the mass transfer equation In the UDF code, the bubble number density value Nb of 4e+09 is chosen, which is discussed in section 4.2.5 The maximum of difference between P* and Psat
(Pa) (dp = Psat – P*), and the accommodation coefficient are set at the values of
75 Pa and 1.2 respectively as recommended in the proposed model
The SST K-ω model with compressible effect is chosen as the turbulent model for the Y+ and Grid convergence index (GCI) tests in section 4.2.2 below.
Boundary conditions and method
The boundary conditions of simulation are listed in Table 13 below:
Wall Stationary wall with no slip condition and standard roughness
To solve the problem, the PRESTO pressure-velocity and second order upwind solutions are applied Finally, the first order implicit is applied for transient formulation, as well as, the time-step is fixed to 1e -5 (s)
Gradient Least squares cell based
Volume fraction Second order upwind
Turbulent kinetic energy Second order upwind
Specific dissipation rate Second order upwind
Transient formulation First order implicit
Simulation results
Solution convergence type identification
After simulating several cases, the results show that the type of convergence of the thesis simulations is periodical The Figure 4.5 presents the trend of scaled residuals of continuity and other parameters having the same fluctuation trend line as the continuity Besides, the fluctuated mass flow rate trend lines of inlet
39 and outlet, pressure and velocity at throat position are also shown in Figure 4.6, Figure 4.7 and Figure 4.8, respectively All extracted result information in the thesis are average values of at least 10 samples at various times to ensure the efficiency and reliability of the simulation result data
Figure 4.6: Mass flow rate trend line chart
Mesh convergence evaluation
To decide the optimum Y+ applied to the final thesis simulations, several Y+ values with meshes are used to simulate and compare results to each other The
41 detailed information of first cell height and the maximum aspect ratio of each mesh is presented in Table 15 below
Table 15: First cell height of mesh information
Mesh First cell height (m) AR Max
The radius velocity profiles in three sections along nozzle described in the Figure 4.9 below are measured to find out the effect of Y+ to velocity profile at these sections and chose the final optimum Y+ value for the thesis
Figure 4.10: Velocity profile comparison at section 2 of mesh 4
In the above velocity profile, the mesh with Y+ value of 100 is much different than the others The velocity profile with Y+ of 35 and 18 have the similar trend lines in all sections with the maximum relative error is about 8% at positions near the center At the remaining locations, the relative error is small and neglectable In addition, the other velocity section profiles also provide the minor difference between the mesh with Y+ value of 35 and 18 Because of the log-law layer of wall functions is applied in the standard K-ω model, the Y+ value between 30 and 300 is acceptable to reduce the mesh sizing and computational domain Therefore, the Y+ value of 35 is chosen as a result The detail information of Y+ comparison in 4 meshes is listed in Appendix D
An important note is that the Y+ equal to 1 and below 5 will not be applied to the thesis because an extremely high density of mesh elements is required to each mesh to improve the mesh quality, especially the aspect ratio parameter,
43 therefore the timestep must be lower than 1e -7 (s) due to the high density With the timestep of 1e -7 (s), the simulated time needs to get the convergence for a simple case is too long, which can be reached to at least of a month
After choosing the optimum Y+ value of 35, the next step is to find out the suitable mesh before doing the turbulent model evaluation Four parameters are used to compare meshes including the average inlet mass flow rate, the average outlet mass flow rate, the minimum throat turbulent kinetic energy, and the outlet average void fraction The outlet average void fraction formula is presented in Eq.(26) below
VF(r = r max) + VF(r = 0) AVG Void Fraction At Outlet = Average
Table 16: Grid convergence index information
Min Throat Turbulent Kinetic Energy (m 2 /s 2 )
The result shown in Table 16 above, the mesh 3 is chosen because of following reasons:
- The difference of the average inlet mass flow rate, as well as the average outlet mass flow rate between the mesh 3 and mesh 4 is below 2%, which is neglectable
- The difference of the average inlet and outlet mass flow rate of the mesh
3 is less than 6%, which is acceptable for the complex “flashing flow” phenomena
- The difference of the minimum throat turbulent kinetic energy between the mesh 3 and mesh 4 is less than 4%
- The difference of the outlet average void fraction between the mesh 3 and mesh 4 is less than 1%, which is neglectable.
Turbulence modeling
A comparison among several turbulence models including Standard K-ε, Realizable K-ε, RNG K-ε, Standard K-ω, SST K-ω is conducted to define the suitable turbulence model for the thesis With each turbulence model K-ε , four wall functions are applied comprised of standard wall function, non-equilibrium wall function, scalable wall function, and enhanced wall treatment The mass flow rate of inlet and outlet in unit of kg/s are used to compare to the experimental mass flow rate of 10.3 kg/s Besides, the pressure along nozzle profiles is also compared to the experiment in Figure 4.11 below The detailed of the turbulence models’ comparisons are presented in Table 17 and Table 18
Figure 4.12: Pressure comparison among K- ω models
Figure 4.13: Velocity contour with the turbulence Standard K- ω (Low-
The simulations of all turbulence models are successful The results show that standard K-ε models provide the most errors of mass flow rate between inlet and experiment, mass flow rate between outlet and experiment, and mass flow rate between inlet and outlet due to the physic of flashing flow phenomena strongly related to separated flow, non-equilibrium phase change effects and the great depressurization that are not the strength of the standard K-ε models The
RNG and Realizable K-ε models are higher versions of standard K-ε models that are added equations for the separated flow and strong adverse pressure gradients to improve the simulation performance However, these models are not suitable because the errors are still really high
The mass flow rate errors of turbulence K-ω models are below 4% that are acceptable, except the SST K-ω model having the mass flow rate error between outlet and experiment about 8% The pressure profiles of the K-ω models are displayed in Figure 4.12 above The standard K-ω model with Low-Re correction option has the closest pressure profile compared to the experimental data at most of points, except the area around throat section from 0.4m to 0.6m in x ordinate with the maximum errors about 15%, in which the greatest bubble
48 growth rate, and the strongest non-equilibrium phase change occur In addition, the standard K-ω model is usually preferred to the internal flows, separated flows, and flows with high pressure gradient, based on recommendations from Ansys
Furthermore, from velocity contour of the standard K-ω model with Low-Re correction option displayed in Figure 4.13 shows that two areas including the inlet area and the one next to the throat section have smaller velocity magnitude values than the others inside nozzle The Low-Re correction option supports to solve these unexpected low-velocity areas and increases the convergence and stability of the simulation
In the above explanations, the standard K-ω model with Low-Re correction option is the most suitable model for the flashing flow simulation and will be chosen as the turbulence model applied in the thesis.
Pressure profile analysis
Pressure is one of the most interested parameters in flashing flow phenomena The complex mixture flow behavior occurs inside the nozzle under non- equilibrium effects, which leads to the unpredictable changes in pressure along nozzle The pressure trendline of the flashing simulation with the chosen turbulence model is presented in Figure 4.14 below
Figure 4.14: Pressure profile with the turbulence Standard K- ω (Low-Re correction) model
The simulated pressure profile at the inlet and outlet shows a good agreement to the experiment data However, the simulated profile from the middle position of constant section to the throat section is slightly different to the experiment with the maximum error value of 15% This is mainly due to the neglect of pressure jump across the interface of two phases, as well as, the limit of mixture model, in which only slip effects between two phases are considered The other interfacial forces such as lift, drag, wall lubrication, and turbulent dispersion force are not taken into account.
The effect of bubble number density on pressure analysis
Bubble number density is one of the most important parameters of flashing flow because it is related to the interfacial density, which is defined to be the ratio of the projected surface area of the bubbles or particles in a cell, during the phase change process Therefore, an analysis of its effect is taken into account in the thesis by changing the bubble number density in UDF code and keeping the rest of the parameters, as well as, boundary conditions The detailed mass flow rate
50 and pressure along nozzle comparisons are presented in Table 19 and Figure 4.15 below
Table 19: Mass flow rate comparison of bubble number density values
Density MFR Inlet %Error MFR Outlet %Error
Figure 4.15: Pressure comparison of bubble number density values
The results show that the inlet and outlet mass flow rates of bubble number density values of 4e+07 and 4e+08 are much different to experiment In addition, the pressure profiles of these bubble number density at all points from the inlet to the throat section is much lower than experimental data The reason is because with the lower bubble number density, the interfacial area density is smaller, which leads to the longer delay period phase of flashing flow, the decrease of bubble growth rate, and the higher value of mass flow rate In the inlet, the fluid can be seen as incompressible, therefore, a greater pressure drop is needed to have the higher mass flow rate in the inlet
51 The pressure continues to decrease to the throat section In this duration, the water fraction is still high, therefore, the incompressible effect of water can be used, which provides an explicit inversely proportional relation between static pressure and velocity (increase of velocity leads to decrease of static pressure and vice versa) After the void fraction increases significantly at throat section (the cue of the evaporation), which is presented in Figure 4.16, the flow becomes compressible and static pressure in this region does not keep decreasing further with the increase of velocity
Finally, the bubble number density value of 4e+09 is chosen because it provides the acceptable mass flow rate errors, as well as, the closest pressure profile compared to the experiment’s pressure chart.
Inlet turbulent intensity analysis
Turbulence boundary conditions at the inlet of the nozzle have been defined by a combination of hydraulic diameter equal to 0.0655 (m) based on the real geometry and turbulence intensity The turbulent intensity is a parameter that affects to the mass flow rate values Therefore, another sensitive case related to the turbulent intensity of inlet varied from 1% to 5% are discussed in Table 20 below During the inlet turbulent intensity analysis, the fixed bubble number density of 4e+09 is applied Besides, the rest of the parameters, as well as, boundary conditions are also kept
The result shows that relationship between the mass flow rate of inlet and turbulent intensity is totally inverse, in general, the lower turbulent intensity will get the higher mass flow rate value The mass flow rate outlet values of the cases applying the inlet turbulent intensity of 1% and 2% are slightly different to the above statement due to the dynamic value variations during collecting data The phase-change model is significantly affected by turbulent kinetic energy; a higher value of turbulent kinetic energy leads to a stronger evaporation process Therefore, the mass flow rate value decreases corresponding to the raise of inlet turbulence intensity
It is found that a turbulent inlet intensity of 4% shows the best accuracy of mass flow rate between inlet with experiment, as well as acceptable error values below 2.6% between the outlet mass flow rate with experiment and between the inlet and outlet For this reason, a turbulence inlet intensity of 4% has been selected for the thesis analysis.
Average void fraction analysis
Another attentive parameter in flashing flow phenomena is void fraction Because the two-phase flow behavior inside nozzle is extremely complicated, void fraction trendline is hard to predict accurately The simulated average void fraction analysis in global and local scopes are conducted successfully, in which the UDF code with chosen bubble number density, the fixed inlet turbulent intensity value of 4% are applied
4.2.7.1.1 Global average void fraction profile
From the experiment report, the process measuring void fraction is very difficult to evaluate and the averaging process in the cross section certainly led to a loss of accuracy, which means that the experimental void fraction profile isn’t really reliable From the simulation, the average void fraction profile in Figure 4.16 below has the same trend line as the experiment, in which, the vapor fraction will be delayed at the begin, and start to increase the rate continuously to the outlet However, the simulated void fraction profile is totally different to the experiment data although the simulation results of mass flow rate and pressure along the nozzle are very close to the experiment data Therefore, the following sections will focus on discussing the simulated average void fraction profiles
Figure 4.16: Average void fraction profile with the turbulence Standard
According to the simulation, the void fraction equals to nearly zero from the inlet to the x - position of around 0.2m, which is near the middle of constant section following the inlet convergence This is the sign of “delay” phase, in which, the surface tension forces are dominated and limit the nuclei formation After passing the “delay” period, the void fraction rate increases slowly at the beginning; it can be seen clearly in the Figure 4.16 where there is nearly constant
54 value range of void fraction rate before it changes suddenly in the x - position of around 0.3m The nearly constant line indicates the first sub-stage of bubble growth time, when the bubble radius exceeds a critical size and the surface tension is still dominated, and the radius of bubble tends to increase slowly The remarkable changed void fraction rate from the x - position of around 0.3m to the throat section points out that the second sub-stage of bubble growth happens, in which the surface tension is weaker, and the bubble radius increases continuously
After flowing into the diverging section, the void fraction continues to increase due to the spontaneous expansion of gas phase behavior whose purpose is to fill up the remaining space of divergence section as much as possible
Figure 4.18: Mixture fluid density contour
With the high void fraction at outlet, the mixture density becomes a noticed parameter affected to the velocity Because of the inverse density and velocity relationship, the velocity continues to increase due to the great effect of the lower density of mixture fluid at the diverging section and outlet extension However, the simulated mixture density value varies in diverging section and outlet extension, therefore, the mixture velocity values in these sections are not constant
Figure 4.20: Average void fraction at nozzle axis
In the Figure 4.17 above, there are some positions at outlet in which the average void fraction decreases The condensation may occur at these positions because
56 of the local pressure recovery A local group of gas particles will be condensed and turned into liquid phase, and vice versa The plot of average void fraction at nozzle axis presented in Figure 4.20 above provides an inconstant void fraction rate at outlet nozzle There are some special points, in which, the rate drops instead of increasing continuously The condensation occurs in these places
4.2.7.1.2 The effect of bubble number density on void fraction
Beside the effect of bubble number density on the pressure presented in section 4.2.5 above, it might also influence to the global void fraction Therefore, a case study of bubble number density effect on the void fraction is performed by changing only the bubble number density values
Figure 4.21: Average void fraction of bubble number density values
The simulation result shows that the fastest increase of void fraction rate occurs with the highest bubble number density value of 4e+09 In addition, the void fraction rate of higher bubble number density is also greater than the lower density With the lower bubble number density, the interfacial area density is smaller, which leads to the longer delay period phase, and the decrease of bubble growth
To analyze the behavior of local radial void fraction, 8 sections are selected to collect data The 6 / 8 sections are focus on the constant pipe and diverging section to get the full characters of radial void fraction during the phase change process
Figure 4.22: Local average void fraction sections
Figure 4.23: Turbulent kinetic energy contour
Figure 4.24: Average void fraction section profiles
58 The delay stage of flashing phenomena occurs in the entire fluid in the radial direction which is presented clearly in the section 1 with the constant zero-value void fraction line
After flowing into the constant pipe section, the void fraction near wall starts to increase, which is displayed in the section 2 However, the void fraction rate decreases towards the nozzle center It is mainly due to the increase of turbulent kinetic energy at the near wall much faster and remarkable than at the nozzle center This result points out an important character of flashing flow which is the relationship between turbulent kinetic energy and bubble growth rate in the local scope The flashing flow tends to occurs firstly at near wall positions, where the turbulent kinetic energy values are higher than other places in the section When the turbulent kinetic energy increases, the molecules in the liquid will vaporize and try to switch to the vapor phase if they have enough energy to break the strong intermolecular bond between water molecules After that, the bubble growth rate develops gradually towards the center, which is shown in the section 3 The reason can be related to the fluid interaction mechanism, the bond energy between water molecules will become weaker when the molecules’ kinetic energy increase
At the throat section (x = 0.5m), almost fluid particles have passed the delay stage, except the elements closed to the nozzle center (r = 0m), in which the void is still equal to zero The void fraction values at near wall positions increase significantly, especially, the void fraction value reaches 0.8 at the closest wall positions
The void fraction at center line starts to increase slowly after the mixture flow goes into the diverging section, which can be seen in the section 5 The long delay period of the flow particles at the centerline is mainly due to the weak effect of turbulent kinetic energy at the nozzle axis from the inlet to the throat section Besides, the spontaneous expansion of gas phase behavior is an important factor leading to the continuous increase of void fraction in the
59 diverging sections The void fraction at the nozzle center increases rapidly in the diverging section and reaches high value of 0.8 at the outlet extension position of 1.1m, which is presented in section 6, 7 and 8.
Discussion
The turbulence model of the standard K-ω model with Low-Re correction option shows the best fit agreement to the Moby-Dick experiment than the others because of its capability to provide the closest mass flow rate to experiment, capture the pressure along nozzle trendline, etc
The bubble number density and inlet turbulent intensity are determined as critical parameters affected directly to mass flow rate, pressure and void fraction The lower bubble number density leads to the greater pressure drop, as well as, longer delay period of flashing flow, therefore, the mass flow rate generally decreases, and vice versa
In general, the mixture fluid might be seen as incompressible before flowing to the throat and the fluid nature can be explained similarly as the incompressible flow behavior, that provides the inversely proportional relation between static pressure and velocity After passing the throat, the fluid becomes compressible due to the high void fraction One of the critical factors that might be used to explain the reason of the velocity increasement in divergent section is the mixture density The velocity continues to increase due to the great effect of the lower density of mixture fluid at the diverging section and outlet extension
Besides, the condensation in divergent and outlet extension sections is also found due to the pressure recovery that makes local groups of gas particles are condensed Therefore, the immediate void rate drops in some positions at these sections are happened as a result
The local void fraction rate of mixture fluid tends to increase toward to the wall in radial direction due to the high turbulent kinetic energy It also means that the
60 delay period in centerline is longer than other locations After passing the delay period, the local void fraction starts to raise significantly
Numerical results conclusion
The thesis has simulated successfully the flashing flow inside nozzle through the Super Moby-Dick nozzle experiment From there, many properties of flashing flow behavior are found:
• The bubble number density and inlet turbulent intensity are two critical parameters affected the mass flow rate, pressure and void fraction of flashing phenomena
• The existence of condensation at some positions in the divergent and extension sections of the nozzle is a specially interested phenomena due to the pressure recovery
• The local void fraction trend development from center to wall, as well as, the longest delay period in positions near the nozzle axis are also discovered and explained carefully.
Limitations
Based on the mixture model theory, the model applied in the thesis eliminates other forces that may affect the “flashing flow” fluid behavior, including interfacial forces such as lift, drag, wall lubrication, and turbulent dispersion force, etc The fixed value of initial bubble radius size is assumed in the CFD software that does not reflect the behavior of bubble size development during flashing
62 Because the thesis working duration is not long, it is not able to create mesh with the Y+ equal to 1 or less than 5, known as the viscous sublayer range and simulate with the smaller timestep than 1e -5 (s).
Further development
The “flashing flow” term is an interested phenomena which is needed to discover more in different conditions There are some development topics that should be considered:
• Simulating other conditions of the Moby-Dick experiment
• Using the Eulerian Multiphase Model approach, in which the other forces are taken into account, to evaluate the simulation more accurately
• Applying the proposed model to other experiments, for instance, the Edward Blowdown Pipe experiment
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Mass Transfer UDF Code
/* UDF to define a simple mass transfer based on Saturation
Temperature The "from" phase is the liquid and the "to" phase is the gas phase */
#define acc_COEFF 1.2 /*accommodation coefficient*/
#define R 8314.34 /*Universal gas constant R (J/kmol-1.K)*/
#define Nb 4e+9 /*Bubble Number Density*/
DEFINE_MASS_TRANSFER(liq_gas_source1,cell,thread,from_index,fro m_species_index, to_index, to_species_index)
{ real m_lg; real m_boil; real dp; real Area_Den;
Thread *liq = THREAD_SUB_THREAD(thread, from_index);
Thread *gas = THREAD_SUB_THREAD(thread, to_index); real p_vapor = 0.; real dp_liq, dp_gas, dp0, source_evap, source_con;
71 Area_Den = pow(6*C_VOF(cell,gas),2.0/3.0)*pow(PI*Nb,1.0/3.0); p_vapor = p_sat + MIN(0.195*C_R(cell,thread)*C_K(cell,thread), 5.0*p_vapor); dp = p_vapor - MAX(0.1,C_P(cell,thread)); dp0=MIN(0.195*C_R(cell,thread)*C_K(cell,thread), 5.0*p_vapor) + MIN(p_vapor-C_P(cell,gas),75.0);
/*printf("\n RhoK %f",MIN(0.195*C_R(cell,thread)*C_K(cell,thread), 5.0*p_vapor));*/ m_lg = 0.; m_cav=0.; m_boil=0.; if(dp > 0.0)
{ m_boil = acc_COEFF*Area_Den*sqrt(M/(2*PI*R*T_sat))*dp0;
} m_lg=m_cav+m_boil; return (m_lg);
Experimental Pressure Data
Experimental Void Fraction Data
Y Plus Comparison Data
Figure 5.1: Velocity profile comparison at section 1 of mesh 1
Figure 5.2: Velocity profile comparison at section 2 of mesh 1
Figure 5.3: Velocity profile comparison at section 3 of mesh 1
Figure 5.4: Velocity profile comparison at section 1 of mesh 2
Figure 5.5: Velocity profile comparison at section 2 of mesh 2
Figure 5.6: Velocity profile comparison at section 3 of mesh 2
Figure 5.7: Velocity profile comparison at section 1 of mesh 3
Figure 5.8: Velocity profile comparison at section 2 of mesh 3
Figure 5.9: Velocity profile comparison at section 3 of mesh 3
Figure 5.10: Velocity profile comparison at section 1 of mesh 4
Figure 5.11: Velocity profile comparison at section 2 of mesh 4
Figure 5.12: Velocity profile comparison at section 3 of mesh 4