CFD Annular Flow Modelling Based on a Three-Field Approach

CFD Annular Flow Modelling Based on a

Three-Field Approach

Erik Skoog

Engineering Physics and Electrical Engineering, master's level

2020

Luleå University of Technology Department of Engineering Sciences and Mathematics

Abstract

This master thesis aim to model the annular flow that occurs in thefinal section between the fuel rods inside Boiling Water Reactors, by ap-proximating the geometry to a cylindrical pipe. Simulations were per-formed in the software ANSYS Fluent, as a step in the development ofreplacing the 1D correlations currently used in the nuclear industry withCFD models in 3D. An Eulerian-Lagrangian approach was used for thethree fields of steam, liquid film and liquid droplets in the model. En-trainment was modeled based on 1D correlations from Okawa [7] anddeposition with the built in Discrete Phase Model in ANSYS Fluent. Thework focused on making the process less time consuming, and increasingaccuracy of the model by comparing the results with empirical data basedon experimental values. A transverse velocity was applied on the dropletsat the point of entrainment with better correlating results with the Okawamodel.

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AcknowledgementsFirst, a big thank you to my primary supervisor Tobias Strömgren. His knowl-edge on CFD has been extremely valuable, and has always been there to answermy questions. Also, thank you to Jean-Marie Le Corre, who has offered hisgreat expertise on the area and provided for the main guidance on the generaldirection of this work. Thank you to Carl Adamsson for his constant knowledgesharing on the area and letting me use his experimental results for compari-son. Thank you to Yann Le Moigne for observing my work and providing withfeedback and technical support throughout the work. Thank you to Anna-LenaLjung, my examinator at Luleå University of Technology.

Thank you to Westinghouse Electric Sweden in Västerås for allowing me toperform my master thesis for the company. Finally, a big thank you to the SEPdepartment at the workplace for welcoming me into the team as one of theirown. It has been a great pleasure working alongside with you and getting toknow you all.

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Contents1 Introduction 7

2 Background 82.1 Boiling Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Properties of Annular Flow . . . . . . . . . . . . . . . . . . . . . 9

3 Theory 103.1 Equations in Fluent . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1.1 Steam Core . . . . . . . . . . . . . . . . . . . . . . . . . . 103.1.2 Liquid Droplets . . . . . . . . . . . . . . . . . . . . . . . . 103.1.3 Liquid Film . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 1D-correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2.1 Deposition . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2.2 Entrainment . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Method 134.1 Overview of previous work . . . . . . . . . . . . . . . . . . . . . . 134.2 Changes and Improvements . . . . . . . . . . . . . . . . . . . . . 15

4.2.1 Liquid Film . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2.2 Automation and Initialization . . . . . . . . . . . . . . . . 164.2.3 Steam and Particles . . . . . . . . . . . . . . . . . . . . . 17

4.3 Post-Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.4 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5 Results 19

6 Conclusions 26

7 Future Work 27

8 Appendix A 28

9 Appendix B 35

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List of Figures1 Image of different flow regimes [11] . . . . . . . . . . . . . . . . . 82 Sketch of the setup of the model. Note that the image is not to

scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Cross sectional mesh of the geometry, containing 1300 cells. . . . 154 Injection wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Deposition rate when injecting particles in two different ways.

G = 1250kg/m2s . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 1D calculation from onset of annular flow using the equilibrium

condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 Total mass flow rate G = 750kg/m2s, compared with experimen-

tal results from the same conditions. . . . . . . . . . . . . . . . . 208 Total mass flow rate G = 1250kg/m2s, compared with experi-

mental results from the same conditions. . . . . . . . . . . . . . . 219 Total mass flow rate G = 1750kg/m2s, compared with experi-

mental results from the same conditions. . . . . . . . . . . . . . . 2110 Film thickness across the perimeter of the pipe. Starting at

z=2.54 m, and slowly decreasing as the film continues furtherdownstream. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

11 Droplet mass flow for different droplet fractions of total amountof liquid applied at the onset of annular flow. G=1250 kg/m2sand steam quality x = 51% at outlet. . . . . . . . . . . . . . . . . 22

12 Mass flow rates for the three fields, G = 750kg/m2s and steamquality at outlet x = 75% . . . . . . . . . . . . . . . . . . . . . . 23



15 Deposition rate divided by concentration of droplets, where theconcentration is determined as in equation (12) with the valuesfor the droplet and steam mass flows obtained from Fluent. Thesolid lines are 1D calculations using the Okawa correlations andcorrespond color wise to the same inlet mass flow as labeled inthe figure. The step wised dotted curves are values obtained fromsimulations in Fluent . . . . . . . . . . . . . . . . . . . . . . . . . 25

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NomenclatureList of Symbols

δ Liquid film thickness, m

m Mass flow rate, kg/s

µ Dynamic viscosity, Pa·s

ρ Density, kg/m3

σ Surface tension, N/m

τ Shear stress, N/m3

D Diameter, m

f Friction factor, dim-less

G Total inlet mass flux, kg/m2s

g Gravitational acceleration, m/s2

J Volumetric flux, m/s

k Mass transfer coefficient, dim-less

p Pressure, Pa

r Radius, m

Sm Mass source term, kg/m2s

Smom Momentum source term, N/m2

T Temperature, K

u Velocity, m/s

Subscripts

d Liquid droplets

dep Deposition

ent Entrainment

f Liquid film

i Interface

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k Liquid or vapor

l Liquid

v Vapor

w Wall

List of AbbreviationsBWR - Boiling Water Reactor

CFD - Computational Fluid Dynamics

DPM - Discrete Phase Model

GUI - Graphical User Interface

LPT - Lagrangian Particle Tracking

TUI - Text User Interface

UDF - User Defined Functions

1D - One Dimension

2D - Two Dimensions

3D - Three Dimensions

6

1 IntroductionIn Boiling Water Reactors (BWRs), liquid water is heated until it vaporizesand become steam, which in turn drives turbines to generate electricity. Heatis transferred from tall cylindrical fuel rods to the water flowing between therods. When this heating occurs, the water passes through various flow regimesresulting in a phenomenon called annular flow at the upper part of the core.This is the final flow regime before all liquid has been vaporized due to theheat flux from the rods. If all liquid film surrounding the rods vaporize beforereaching the upper end of the core, dryout is inevitable. This must be avoided,since dryout can cause severe damage on the equipment due to the less efficientheat exchange occurring between the rods and the steam when the liquid filmvanishes. Thus it is essential for companies in the nuclear industry to have well-functioning and accurate models that can predict the critical heat flux whendryout occurs.

Researchers Haipeng Li and Henryk Anglart at KTH in Stockholm has inclose cooperation with Westinghouse Electric Sweden developed a computa-tional fluid dynamics (CFD) model for this purpose using the open source soft-ware OpenFOAM. Following that, the previous master thesis student MariaCamacho worked on improving the model and implemented it in the softwareANSYS Fluent [4]. The following year, the master thesis student SalvatoreRaddino continued the work of Camacho and made several improvements tothe model; increasing accuracy and decreasing computational time [8]. Thismaster thesis is a continuation of the work previously performed by Raddinoand Camacho on the modelling of annular flow using CFD based on a three-fieldapproach.

During the CFD-modeling of annular flow in cylindrical pipes, three mainareas are considered; the steam core occupying the center of the pipe, liquidfilm surrounding the pipe along the walls, and liquid droplets within the steamcore. These three fields have three main ways of interacting with each other.Entrainment is the rate at which droplets are formed and released from thefilm into the steam core, deposition is the rate at which liquid droplets aredepositing on the liquid film and vaporization from the liquid film to the steamcore. Vaporization is dependent on the power distribution along the rods. Inthis work, only uniform power distribution is considered.

Most of the current studies on annular flow are based on empirical corre-lations in one dimension along the pipe. The disadvantage in relying on thesecorrelations mainly consist of the severe restrictions in the use of different pa-rameters. Being able to perform meticulous CFD-simulations in 3D will allowfor a more detailed understanding of all complex events occurring in the flow.

In 2003, Okawa et al. developed 1D-correlations for the rates of entrainmentand deposition in perfectly cylindrical shaped pipe geometries[7]. The CFD-model in this thesis will make use of the entrainment correlation from Okawa,whereas deposition will be determined by calculating particle trajectories witha Lagrangian approach, using the Discrete Phase Model (DPM) provided inANSYS Fluent, the software that will be used for the simulations. The purpose

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with this is to obtain an understanding for how sensitive the model is to thisway of modeling deposition, and results will be compared with 1D-calculationsbased on the Okawa correlations and experimental results.

The main objective of this thesis will be to study the sensitivity of the modelto different inlet mass flows with an implemented droplet transverse velocityat the point of entrainment. At the onset of annular flow, different fractionsbetween droplets and liquid film will be considered. Droplet- and film mass flowwill then be simulated based on these different fractions at the onset of annularflow. These simulation results will then be compared with calculations basedon the Okawa correlations.

Another objective will be to study the injection of droplets and how a realisticparticle injection affects deposition.

2 Background

2.1 Boiling RegimesThere are a few flow regimes for the fluid to pass through before reaching annularflow and eventually completely evaporate. Bubbly flow, slug flow, churn flowand finally annular flow, all seen in figure 1. Bubbly flow occurs as soon assmall vapor bubbles start to appear in the flow. As these bubbles increase insize and more liquid is vaporized, large slugs start to appear. As the fraction ofsteam increases, so does velocity, since pressure is kept constant in the reactorcore. The slugs will later be torn apart into smaller pockets of steam as velocityis constantly increasing, causing the transition to churn flow. With furtherincreasing velocity, the forces acting upon the pockets filled with vapor willbecome larger and make them unstable.

Figure 1: Image of different flow regimes [11]

Finally, annular flow with a liquid film circumfering the walls of the pipeand liquid droplets inside the steam core is formed. All simulations performedin this master thesis will only consider annular flow.

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The physics behind the transitions between these flow regimes is extremelycomplex. Based on experimental data, G. B. Wallis provides the following cor-relation as a condition where the transition between churn and annular flowoccurs [10],

J∗v = 0.4 + 0.6J∗

l . (1)

The dimensionless volumetric fluxes for the vapor and liquid J∗k are each defined

asJ∗k = Jk

√ρk

gD (ρl − ρv). (2)

Here, g is the gravitational constant, D the pipe diameter and ρ the density.The subscript k denotes the phase in question, liquid or vapor. This method todetermine the onset of annular flow was used by Okawa during the developmentof the 1D-correlations of deposition and entrainment [7]. It is therefore of utmostimportance to use the same method in this work, since those 1D-correlationswill be studied.

2.2 Properties of Annular FlowThe majority of the volume in the pipe is occupied by a steam core locatedin the center. The steam core mass flow is continually increasing due to themass transfer occurring through evaporation of the liquid film. The rate ofevaporation is determined by the heat flux from the wall and its distribution.

The liquid phase is divided into droplets and film. Liquid droplets inside thesteam core are heavily affected by the turbulence forces in the vapor. This isparticularly true for smaller droplets, since larger droplets carry more mass andthus possess a larger momentum. Droplets inside the steam core depositing onthe liquid film surface are absorbed by the film, thus adding mass to it. Therate of deposition is directly proportional to the concentration of droplets in thesteam core.

On the vapor-liquid interface, disturbance waves are nearly always present.According to P. B. Whalley, droplets entrain from the film if the liquid filmwaves reach a certain height [12]. Experiments on this phenomenon wherephotos have been taken cross section wise, show that for large enough Reynoldsnumbers, the wave amplitudes are high enough for the liquid film to shatter intothe steam core, quickly splitting up into several smaller droplets. This is calledentrainment, and is directly proportional to the film thickness. The Reynoldsnumber is in this case determined by

Ref =ρfJfD

µl(3)

where µl the liquid dynamic viscosity.In this thesis, the mass transferring through vaporization, deposition and

entrainment are considered and modeled.

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3 Theory

3.1 Equations in Fluent3.1.1 Steam Core

The steam is treated as a single-phase flow in Fluent and a two-way couplingis imposed on the steam and droplets. The continuity equation (4) is solved,containing a time derivative and a divergence term, describing the change ofmass in time and space,

∂ρv∂t

+∇ · (ρvuv) = Svap(z). (4)

The source term on the right-hand side corresponds to the mass transfer thatis constantly occurring from the film to the vapor through vaporization. Here,z denotes the axial direction of the pipe.

The source term is defined as the fraction between the power heat flux en-tering the pipe from the wall and the latent heat of vaporization described bythe difference between the saturated enthalpies of vapor (hv) and liquid (hl),

Svap(z) =q(z)

hv − hl. (5)

The power heat flux q(z), can in theory possess any distribution along the pipe.Fluent also solves the momentum conservation equation for the vapor phase,

seen in equation (6),

∂(ρvuv)

∂t+∇ · (ρvuvuv) = −∇pv + ρvg+∇ · τeffv + Smom,vap + Smom,drop. (6)

Here, acceleration through the time derivative and divergence term is describedon the left-hand side. On the right-hand side are pressure, gravity and stressforces along with two momentum source terms from vaporization of the film(Smom,vap) and droplet flow in the steam core (Smom,drop).

3.1.2 Liquid Droplets

Calculations on the particles will be performed in Fluent using the built-inDiscrete Phase Model (DPM). Instead of performing calculations on each in-jected particle, Fluent treats the particles as part of parcels using an Eulerian-Lagrangian approach. A parcel can contain several particles with similar proper-ties and makes the calculations in Fluent faster. Several parcel release methodscan be chosen in Fluent. When injecting droplets with a User Defined Function(UDF), the parcel release method is automatically set to standard and cannotbe changed. The standard release method injects a single parcel per injectionstream per time step. The number of particles per parcel is determined as

NP = ms∆t

md(7)

10

where NP is the number of particles in a parcel, ms is the mass flow rate of eachparticle stream, ∆t is the time step and md is the particle mass. ms can becalculated by taking the total particle mass flow divided by the number of facetsin the cross sectional mesh. With a ∆t of 0.0005 seconds, a particle diameterof 1.2mm, and a total mass flow rate of 1250kg/m2s, the number of particlesper parcel is 0.0585. Since every facet in the cross sectional mesh is subject toone injection stream and thus one parcel at every time step, this number can bebelow one. The simulation needs to be running over a sufficient amount of timesteps for enough parcels to be injected and hence create a statistically accurateresult for the particles. The particle trajectories are calculated and tracked untilthey hit the wall and are absorbed by the film, or reach the outlet.

The calculations between the vapor and the droplets are two-way coupled;the change in momentum between these two phases goes both ways. The mo-mentum transfer in this model is computed as

Smom,drop =

(18µCDRe

ρdd2d24(ud − uv) + Fother

)md∆t (8)

where Fother accounts for gravity and drag forces [3] and is calculated as

Fother = g

(1− ρv

ρd

). (9)

3.1.3 Liquid Film

For the modeling of the liquid film, the built-in function Eulerian Wall Films(EWF) is used. The function makes use of the assumption that the liquid filmis thin enough for certain properties to remain constant in the film, independentof radial position. Fluent solves the conservation of film momentum,

∂ρfδuf∂t

+∇s · (ρfδufuf ) = −δ∇spf + ρfδg +3

2τvf −

3µfδµf+

Sm,dep − Sm,ent − Sm,vap (10)

which has certain similarities with the single-phase momentum equation 6. Thefilm thickness is δ, and the third and fourth term on the right hand side ofequation (10) correspond to the net viscous shear force in the vapor-film andwall-film interfaces [3]. Using the EWF, the liquid film is calculated in 2D alongthe wall and hence does not occupy any space in the 3D model.

3.2 1D-correlationsIt is the calculation in 1D that will provide for almost all boundary conditionsnecessary for the simulations. One of these is the fraction of liquid film, dropletsand steam at the inlet. The total mass flow rate can then be multiplied witheach fraction respectively to obtain the mass flow rate for each field. The frac-tion between vapor and liquid at the onset of annular flow can be calculated

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based on the correlations provided in equations (1) and (2). To obtain thisfraction, Utsuno and Kaminaga suggest an equilibrium condition of depositionand entrainment [9]. The fraction is step wised increased from 0 to 1 and en-trainment and deposition are calculated for every step. The fraction betweenliquid droplets and liquid film at the onset of annular flow is then found whereentrainment and deposition are equal.

3.2.1 Deposition

The simulations from Fluent will be compared with 1D-calculations based onOkawa’s correlations of entrainment and deposition. For deposition, the corre-lation used in the 1D-calculation was presented by Okawa et al. [7], based onempirical studies:

Sm,dep = 0.0632

√Cσ

D(11)

where C is the concentration of droplets in the steam core. With the assumptionthat the relative velocity between the droplets and the steam core is negligible,the droplet concentration becomes

C =md

(md/ρl) + (mv/ρv)(12)

as described by Le Corre and Adamsson [6].

3.2.2 Entrainment

Entrainment is calculated as described by Okawa et al. [7], based on empiri-cal studies. This takes into consideration that entrainment only occurs if theReynolds number of the liquid film is above a certain critical value Refc. Thecorrelation provided by Okawa can be seen in equation (13) and is used in theconservation of mass calculations of the liquid film in equation (10).

Sm,ent = keρlfiρvJ

2v δ

σ

(ρlρv

)nfor Ref > Refc (13)

where the subscripts l and f denotes the liquid phase and the liquid film, respec-tively. ke and n are constants with recommended values of 4.79 · 10−4m/s and0.111 respectively. The superficial vapor velocity is denoted Jv and the criticalfilm Reynolds number is set to Refc = 320. The interfacial friction factor fi isseen in equation (14) and is dependent on the film thickness, which is obtaineddirectly from Fluent,

fi = 0.005

(1 + 300

δ

D

). (14)

In the 1D correlations the film thickness is estimated from the force balancebetween the interfacial shear force and the wall friction force, both acting onthe liquid film. It is described as

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δ =1

4

√fwρlfiρv

JfJvD (15)

where the wall friction factor fw is recommended by Wallis [10] to be calculatedas

fw = max(

16

Ref, 0.005

). (16)

Equations 13 and 14 are implemented in the UDF to control the film masssource. All constants are defined in advance and the Reynolds number and filmthickness are calculated by Fluent.

4 Method

4.1 Overview of previous workThis master thesis is largely based on the two previous thesis works performed byMaria Camacho and Salvatore Raddino. Camacho developed the first model inFluent and compared it with the results obtained from an OpenFOAMmodel de-veloped at KTH. Raddino continued the work of Camacho and implemented gen-eral improvements to the model, with more accurate results and vastly shortercalculation time. The model developed by Raddino contains two simulationregions. One inlet development model and one model for the main simulation.The reason for this setup is to provide enough time for the fluid to fully developinto annular flow. Each of these models consist of two regions. First, the 1.06meter long annular_wall, where simulation data is obtained and comparedwith theoretical results. This corresponds to the last section of the in total 3.65meter tall fuel bundles. Next, a shorter injection_wall (0.1103 m) just beforethe previously mentioned region. From here the liquid film is injected radiallyand a vapor velocity is defined at the inlet. A sketch of where these two regionsare found in the setup can be seen in figure 2.

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Figure 2: Sketch of the setup of the model. Note that the image is not to scale.

In both previous master theses, the modeled deposition rate can be observedas either a substantial underestimation (Camacho) or an intense peak (Raddino)at the very first part of the simulated pipe. This is the result of an entrance ef-fect, created by the defined inlet conditions for the droplets. Steam and dropletsare injected at the inlet of the first model with a uniform velocity profile. Thiscreates indeed an entrance effect, although this is irrelevant since only the outletof this simulation is of interest. For the steam, velocity properties are saved atthe outlet of the first model as a profile file and uploaded as initial conditionfor the steam for the main model. The particles are treated in the same way,where the properties of the particles are saved in an injection file. This file isthen defining the initial conditions for the particles in the main model, wheredroplets are injected at the inlet of the annular_wall region, using the DPM.

One of the changes made by Raddino regarding the particles was the im-plementation of injection files. Instead of injecting the particles uniformly overthe inlet surface of the main model, particles are injected through an injection

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file containing all necessary information of the particles. Particles are first in-jected uniformly through the inlet surface of the InletDevelopment model anddevelop a more realistic distribution at the outlet during that distance. Thepositions and velocities of the particles are then saved in an injection file, whichis implemented at the inlet of the main model. Seen in figure 3 is the cross sec-tional mesh used in this work. This mesh has not undergone any changes sincethe first OpenFOAM model constructed by Li and Anglart which, according toCamacho, also managed to confirm grid independence for this mesh. In orderto better capture the small local changes that might arise in the flow closer tothe wall, the mesh is more refined in this area, containing a higher number ofelements in the radial direction closer to the wall.

Figure 3: Cross sectional mesh of the geometry, containing 1300 cells.

4.2 Changes and Improvements4.2.1 Liquid Film

An overall script was constructed, with the purpose of being well synchronizedwith the models in Fluent. The script, provided in Appendix A, would per-form 1D-calculations on various parameters of the flow, such as entrainment,deposition and film thickness based on correlations constructed by Okawa et al[7]. The results from these 1D-calculations would then be used as initial andboundary conditions for the model in ANSYS Fluent. The necessary conditionsinvolving the three fields are flow fluxes of droplets and film, and vapor velocity.Conditions regarding the surrounding environment include pressure, tempera-ture, heat flux and power distribution. It is based on these latter parameterswhich more specifically determines the environment, constants such as densityand viscosity of the two phases are defined.

The results from the simulation would then be compared with the calcula-tions from the 1D-model, and hopefully correlate rather well. An issue with the

15

model was the mismatch in the manually calculated input value for the liquidfilm flow and the simulated film flow right at the start of annular_wall.

Figure 4: Injection wall

This problem was solved by splitting the first region seen in figure 4 into twoparts (A and B) and inject the liquid film only through part B. More liquid filmwould then need to be injected, since the injection area would shrink with 1/9.

One of the more obscure boundary conditions is the "z-momentum flux". Aforce that is applied in the injection_wall, setting the liquid film in motion. Inthis work, it was determined through trial and error by comparing the simulatedfilm thickness and the theoretical film thickness at the inlet. A larger forcewould result in a higher initial film velocity, decreasing the film thickness. Asmaller force would have the opposite effect, causing an increase of the initialfilm thickness. For G = 1250kg/m2s, it was approximated to 55.7N/m2.

4.2.2 Automation and Initialization

There are two ways of using ANSYS Fluent; by using the graphical user interface(GUI) or the text user interface (TUI) in the command window. One majorprogress from the previous work by Raddino, has been the automation of thesimulations with the use of journal files. A journal file is a file containinginstructions in text format, which are executed by ANSYS Fluent line by line.By implementing all conditions needed for a specific simulation into a journalfile, Fluent can be ran entirely without graphics, only making use of the TUI.

This modification makes it much less time consuming to run several sim-ulations with different boundary conditions by completely avoiding using theGUI. The computational time required to perform one simulation varies greatlydepending on the implemented boundary conditions, but for the three mass flowcases (G = 750kg/m2s, G = 1250kg/m2s and G = 1750kg/m2s) the computa-tional time decreased from approximately five hours to just below one hour.

Based on studies of several simulations on the matter, it is estimated thatthe flow is fully developed at 0.325 seconds. That is, after 650 time steps. Withthis in mind, it is only desirable to export solution data after 650 time steps.

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From there, solution data is exported in ASCII-files at every fifth time step until1000 time steps (0.5 seconds) are reached. This gives 70 ASCII-files to averageover, and is an effective way of reducing noise from specific time steps. The filesare saved in the same folder as a Matlab script used for reading the files andplotting the result.

Along with this process, an improvement made for the particle injection wasthe removal of the need of reformatting the file using a separate C script.

4.2.3 Steam and Particles

The first InletDevelopment model was seen as a redundant element of the sim-ulation process. With the existing setup, in order to run a simulation withcertain conditions, the first InletDevelopment model had to be initialized andcompleted first. For its removal, the steam and particles had to be injectedin a sufficiently realistic way, without relying on the first model to generatereasonable particle and velocity profiles.

For the particles, two types of injections are implemented. One at the inletusing a particle injection file, and particles injected into the core from the wallthrough entrainment using a surface injection. For the injection at the inlet,various injections were tried and compared with the results from the Okawa1D-model. Keeping one set of original particle velocities (first obtained byrunning the InletDevelopment model), and multiplying their transverse velocitycomponents in the XY-plane with 0.55 significantly reduced the first peak indeposition as seen in figure 5.

2.4 2.6 2.8 3 3.2 3.4 3.6 3.8

z [m]

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Mass flu

x [kg/m

2s]

Deposition

New injection

Old injection

1D Okawa model

Figure 5: Deposition rate when injecting particles in two different ways. G =1250kg/m2s

17

As for their axial velocities, keeping an original distribution but varying themean axial velocity proportional to 95% of the injected steam velocity providedthe most accurate results. The droplet mass flow was calculated manually andmultiplied with each cell area in the injection surface to create a uniform in-jection of droplet mass flow. This also proved to be a more accurate method,rather than applying a droplet mass flow with a certain distribution. Looking atfigure 5, it is interesting to notice the slope of the curve from the new injectionat z=3.1. Although, it is slightly overpredicting the Okawa correlation, it seemsto decrease at a rather similar rate as the correlation under these conditions.

In the previous model, newly formed droplets detached from the film weregiven the same velocities in all directions as the steam in that specific positionof entrainment. A more realistic way of modeling this would involve assigningthese droplets a transverse velocity at the point of entrainment. P. Andreussiand B. J. Azzopardi provide a correlation for this purpose [2], that has been usedby Haipeng Li in previous research. The particle transverse velocity correlateswith

ud = 11.1uτ

√ρvρl

(17)

where

uτ =

√ff · u2free

2(18)

ff is a friction factor determined by

ff = 0.045

(ufreeδ

ν

)−1/4

(19)

and

ufree = uv8

7(20)

where uv is the mean vapor velocity at the current the cross section. Thetransverse velocity would be calculated and applied in the UDF with X- andY-components, assuming a radial direction towards the center of the pipe of thisvelocity. In this work, only 60% of this calculated transverse particle velocityud was applied, due to convergence issues for larger velocity magnitudes. AllUDFs used in this work are provided in Appendix B.

In the previous model, steam was injected as a velocity profile obtained atthe outlet of the InletDevelopment model. This made the profile fixed, impos-sible to adjust the velocity magnitude for different conditions. Injecting steamuniformly, only possessing a velocity in the axial direction proved to not changethe final results noticeably. The steam would be injected at the first inlet of thepipe, passing through the injection_wall region.

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4.3 Post-ProcessingMuch of the focus in this work has been put on making the results obtainedthrough post-processing more reliable. Deposition rate is now obtained directlyfrom Fluent and not manually calculated using a mass balance equation as inthe previous work. During the simulation, ASCII-files containing the desireddata are saved at every fifth time step. The data is then summed and averagedover the time steps where the flow is fully developed.

Since the liquid film is calculated in 2D on the walls, the entire cross sectionalarea can be assumed to be filled with droplets and vapor. By creating crosssectional planes through the pipe, the vapor velocities at these locations can beused to calculate the vapor flow rate. This is done by multiplying the meanvapor velocity with the vapor density and the cross sectional area. The dropletmass flow rate is calculated as a mass balance, by subtracting the vapor andfilm flow rates from the total injected mass flow rate.

4.4 AssumptionsAssumptions have been made based on assumptions documented in previouswork regarding annular flow and the accuracy of the model used for this study.

No vaporization is assumed to occur of the liquid droplets to the steam core.That is, all droplets are tracked by the Lagrangian Particle Tracker (LPT) untilthey deposit on the film or reach the outlet. All droplets are also assumed to beperfectly spherical, and possess a diameter of 1.2 mm. In the previous masterthesis work, Raddino performed some studies on the droplet diameter and 1.2mm was considered to be rather optimal. Studies on droplet diameter have alsobeen performed by Le Corre et al., who concluded that particles in annular flowregimes like these with a diameter of 1.2 mm contain the majority of the totaldroplet mass [5]. For the simulation, droplets are also assumed to not mergewith each other and form bigger droplets, as well as not splitting up into smallerdroplets. This phenomenon is ignored because of the current absence of such acalculation model within the field of CFD.

Saturation conditions are assumed to be present throughout the entire flowregion, and all heat flux on the wall is used for the evaporation of the liquidfilm. All simulations are performed under the condition of 10 K subcoolingand a pressure of 70 bar. Based on these assumptions, a uniform pressure andtemperature can be considered across the entire pipe.

5 ResultsBefore initiating a simulation the 1D-script containing the correlations by Okawaperforms calculations on the specific case. First, the position where the onsetof annular flow occurs is determined. Based on that result, the boundary con-ditions can be obtained for the simulation model. Figures 6a and 6b show theresults from the script using a total flow rate of G = 1250kg/m2s with a steamquality of x = 51% at the outlet and uniform power distribution. Under these

19

conditions, the onset of annular flow appear at z = 1.13 m and the three fieldsalong with the rates of deposition and entrainment can be calculated and plottedonward from that point.

1 1.5 2 2.5 3 3.5 4

z [m]

0

0.02

0.04

0.06

0.08

0.1

0.12

Ma

ss f

low

ra

tes [

kg

/s]

Mass flow rates

Liquid film

Vapor

Droplets

(a) Mass flow rates for the three fields

1 1.5 2 2.5 3 3.5 4

z [m]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Ma

ss f

lux [kg

/m2s]

Deposition and Entrainment rates

Deposition

Entrainment

(b) Deposition and entrainment rates

Figure 6: 1D calculation from onset of annular flow using the equilibrium con-dition.

Experimental results in figures 7-9 have been taken from Adamsson andAnglart [1]. For all three cases, the deposition rate recover from the under-prediction at z = 3.1 and transitions into an overprediction in figures 8 and 9.However disregarding that, the slope of the deposition rate for all cases coincidefairly well with the 1D Okawa model after about 3.1 m. This pattern is observ-able for the film flow rates from Fluent as well, most notably in figures 8b and9b where flow rate initially decreases but start to recover at z = 3.1.

2.4 2.6 2.8 3 3.2 3.4 3.6 3.8

z [m]

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Mass flu

x [kg/m

2s]

Deposition, G=750 kg/m2s

From Fluent

1D Okawa model

(a) Deposition rate

2.4 2.6 2.8 3 3.2 3.4 3.6 3.8

z [m]

0

0.005

0.01

0.015

0.02

Ma

ss f

low

[kg

/s]

Film Flow Rate, G=750 kg/m2s

From Fluent

1D Okawa model

Experimental results

(b) Film flow rate

Figure 7: Total mass flow rate G = 750kg/m2s, compared with experimentalresults from the same conditions.

20

2.4 2.6 2.8 3 3.2 3.4 3.6 3.8

z [m]

0.1

0.2

0.3

0.4

0.5

0.6

0.7M

ass flu

x [kg/m

2s]


From Fluent

1D Okawa model

(a) Deposition rate

2.4 2.6 2.8 3 3.2 3.4 3.6 3.8

z [m]

0

0.005

0.01

0.015

0.02

0.025

0.03

Ma

ss f

low

[kg

/s]


From Fluent

1D Okawa model


(b) Film flow rate


2.4 2.6 2.8 3 3.2 3.4 3.6 3.8

z [m]

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Mass flu

x [kg/m

2s]


From Fluent

1D Okawa model

(a) Deposition rate

2.4 2.6 2.8 3 3.2 3.4 3.6 3.8

z [m]

0

0.005

0.01

0.015

0.02

0.025

Ma

ss f

low

[kg

/s]


From Fluent

1D Okawa model


(b) Film flow rate


Figure 10 shows the film thickness along the wall of the pipe, averagedover several time steps where the flow is fully developed. Small variations areexpected because of the constant deposition of droplets causing changes in thefilm thickness on a local level. That is probably the explanation for the suddenpeak observable at approximately 25 degrees. Droplets are injected just atz=2.54 but do not collide with the wall instantly. Hence, variations can be seen6.1 cm further downstream when the first set of droplets has deposited and noton the blue curve where the droplets are first injected.

21

-200 -150 -100 -50 0 50 100 150 200

degrees

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Film

thic

kness [m

]

10-4 Film thickness across perimeter, 6.1 cm spaced

Figure 10: Film thickness across the perimeter of the pipe. Starting at z=2.54m, and slowly decreasing as the film continues further downstream.

Figure 11 shows the development of droplet mass flow for various conditionsapplied at onset of annular flow. Instead of imposing an equilibrium conditionbetween entrainment and deposition, the fraction between liquid film and liquiddroplets is controlled manually, while the onset of annular flow is still assumedto occur at the same position (z=1.13). When the equilibrium condition isactive, the fraction of droplets is 60.3%. In this case, the power is uniformlydistributed, G=1250 kg/m2s and steam quality x = 51% at outlet.

2.4 2.6 2.8 3 3.2 3.4 3.6 3.8

z [m]

0.085

0.09

0.095

0.1

0.105

0.11

0.115

0.12

Ma

ss f

low

[kg

/s]

Mass flow rate

30% droplets

40% droplets

50% droplets

60% droplets

70% droplets

80% droplets

90% droplets

(a) Simulated results

2.4 2.6 2.8 3 3.2 3.4 3.6 3.8

z [m]

0.085

0.09

0.095

0.1

0.105

0.11

0.115

0.12

Ma

ss f

low

[kg

/s]

1D Okawa model

30% droplets

40% droplets

50% droplets

60% droplets

70% droplets

80% droplets

90% droplets

(b) 1D-Okawa model

Figure 11: Droplet mass flow for different droplet fractions of total amount ofliquid applied at the onset of annular flow. G=1250 kg/m2s and steam qualityx = 51% at outlet.

22

Figures 12 to 14 show the mass flow for all three fields present in the pipe fordifferent total inlet mass flows compared with calculations in 1D based on theOkawa correlations. The field of each 1D curve is corresponding to the curvefrom Fluent it is closest to. The vapor seems to coincide with the Okawa correla-tion fairly well for all cases, but for the liquid film and droplets slight differencesare noticed, specifically in figure 12. The droplet mass flow is calculated as amass balance from the vapor and film flow rates, so it is not unexpected with alarger difference in droplet flow rate when such is noticed in the film flow rate,as in figure 12.

2.4 2.6 2.8 3 3.2 3.4 3.6 3.8

z [m]

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Mass flo

w r

ate

[kg/s

]

Mass flow rates, G=750 kg/m2s

Droplets from Fluent

Liquid film from Fluent

Vapor from Fluent

1D Okawa

Figure 12: Mass flow rates for the three fields, G = 750kg/m2s and steamquality at outlet x = 75%

23

2.4 2.6 2.8 3 3.2 3.4 3.6 3.8

z [m]

0

0.02

0.04

0.06

0.08

0.1

0.12

Mass flo

w r

ate

[kg/s

]




Vapor from Fluent

1D Okawa


2.4 2.6 2.8 3 3.2 3.4 3.6 3.8

z [m]

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Mass flo

w r

ate

[kg/s

]




Vapor from Fluent

1D Okawa


Since deposition is strictly dependent on the concentration of droplets, di-viding the deposition rate with the concentration of droplets can provide some

24

indications for the deposition sensitivity. Seen in figure 15 is the fraction betweendeposition rate and concentration of droplets. The cases where G = 750kg/m2sand G = 1250kg/m2s follow their corresponding 1D Okawa correlation ratherwell. For the case where G = 750kg/m2s, the liquid film has completely vapor-ized at z = 3.4 as can be seen in figure 7. Since the CFD-model is only validfor as long as the liquid film is still present, the results beyond this point arenot trustworthy and have thus been removed. The case where G = 1750kg/m2sis clearly below its corresponding 1D correlation, indicating that the model hasdifficulties in performing calculations on the droplets for this inlet mass flow.As can be seen in figure 9a, deposition differed the most from the Okawa cor-relation for this inlet mass flux, compared to the other cases in figures 7a and8a.

2.4 2.6 2.8 3 3.2 3.4 3.6 3.8

z [m]

0

0.005

0.01

0.015

0.02

0.025

[m/s

]

Deposition over droplet concentration

G=750 kg/m2s

G=1250 kg/m2s

G=1750 kg/m2s

Figure 15: Deposition rate divided by concentration of droplets, where theconcentration is determined as in equation (12) with the values for the dropletand steam mass flows obtained from Fluent. The solid lines are 1D calculationsusing the Okawa correlations and correspond color wise to the same inlet massflow as labeled in the figure. The step wised dotted curves are values obtainedfrom simulations in Fluent

25

6 ConclusionsIt is clear that the LPT in the CFD-model is highly sensitive to changes inproperties of the particles. Many different properties of the injected particleswere tried and rate of deposition was always highly unpredictable. In order toobtain an initial velocity distribution of the droplets, particles were first injectedthrough a surface injection with uniform velocity in axial direction at the inlet.The positions and velocities of all particles in every facet at the outlet were thensaved. This way, the particles would already have an initial realistic velocitydistribution axially as well as radially.

Keeping the velocity distribution in axial direction, although multiplying themagnitude with 90% of the inlet steam velocity provided satisfying results fordeposition rates. The particles would also have an initial transverse velocity inthe (cross sectional) XY-plane. Keeping this velocity unchanged resulted in alarge initial spike in deposition. This can be explained by droplets having a largetransverse velocity depositing on the walls right after the droplets are injected.With a reduced initial transverse velocity, this spike could be reduced. Theinitial transverse velocity was set to 55% of the transverse velocity the particlesoriginally obtained at the outlet of the InletDevelopment model.

The implementation of a transverse velocity for entrained droplets con-structed by [2] proved to generate results more accurate with the Okawa correla-tions for the deposition rate, despite only using a 60% fraction of the suggestedvelocity. Even though it could not be noticed for the first part of the simulation,the deposition rate would be much closer to the Okawa correlation during thesecond half of the geometry. Figure 11 shows an interesting pattern, where thesimulation results seem to follow a somewhat similar shape as the calculated re-sults. Seen in figure 11a, the different fractions seem to merge together furtherdownstream, just as in the 1D-calculation in figure 11b. This is yet anotherindication that the model treats the droplet mass flow moderately accurate.

This master thesis has achieved in making the simulation process and post-process more efficient and less time consuming. An overall 1D script with at-tached functions was constructed based on the Okawa correlations for calcu-lating the boundary conditions and compare the simulation results with. Thepossibility to automatically initiate simulations in Fluent through journal fileswas implemented, using all necessary boundary conditions obtained from the1D script. The "InletDevelopment"-model previously used for obtaining initialvelocity distributions for the droplets and vapor was removed, without decreas-ing accuracy in the final results. By using the TUI through journal files insteadof the GUI, computational time decreased in many cases from several hours tojust below one hour. Based on this, it can be determined that merely constantlyupdating the GUI requires a considerable amount of time for Fluent.

The error in the amount of initial film injected not matching the amountmanually calculated was fixed. The post-processing has been made more efficientand more trustworthy since the data from Fluent is now averaged over severaltime steps, reducing noise from individual time steps.

26

7 Future WorkSince this model has been constructed based on a very basic geometry, thereis much to unravel by developing the geometry to more complex examples. Nodoubt, it would be desirable to reform the geometry to perfectly match flowchannels between the fuel rods in the BWR core. In order to do so, the sim-ulations cannot rely on correlations in 1D, performed on other geometries. Inthis work, entrainment is based on the Okawa correlation whereas depositionis modeled using LPT. Replacing the current entrainment correlation is there-fore absolutely necessary for further studies on other geometries than perfectlyshaped cylindrical pipes. Reducing the sensitivity for deposition would also bea major step towards an accurate 3D model.

As previously mentioned, the transverse velocity for entrained droplets pro-vided by [2] was a success for the cases tried in this work. Further studies onthis matter is needed in order to validate its functionality in the model. Futurework on the matter should also investigate what might cause the convergenceerrors appearing for transverse velocities above 60% of the calculated value.

Knowing the relationship between the boundary condition "z-momentumflux" and the initial film thickness would be advantageous for future simulationswith this model. Since entrainment is directly proportional to film thickness, itwould allow for a more accurate initial prediction of entrainment as well.

The suggestion of implementing an unstructured mesh was first proposed byRaddino. This would require major changes in the UDF, since its construction isbased on the structured mesh currently used. It would also work as another toolfor investigating the mesh dependency of the model. Making a transition froma structured to unstructured mesh paves the way for easier future developmentof more complex geometries. A first change in geometry would be an extensionof the pipe in order to perform simulations from the onset of annular flow.

27

8 Appendix AIn order to implement all boundary conditions correctly, this 1D script withattached functions based on the Okawa correlations were created. To calculateall properties of the saturated water, steam table functions in "XSteam" wereused, provided by Magnus Holmgren. The file "bc.inp" contains most if theinputs and are read in by the script at the start, defining several of the presentconditions. The first script in this appendix is the major 1D script used forcalculating the boundary conditions. Below, is first the function for determiningthe heat flux based on the outlet steam quality. Secondly, the function todetermine the fraction between droplets and liquid film at the onset of annularflow based on the equilibrium conditions. Next is the function to determinethe film thickness and finally, the functions for determining entrainment anddeposition according to the correlations developed by Okawa.

1 close all23 inputf=’bc.inp’;4 bc=readInputFile(inputf);56 % List of available power distributions7 inlet = @(z) 0.01811∗z.^6−0.23174∗z.^5+1.179∗z.^4−2.878∗z.^3+2.8861∗z

.^2−0.11836∗z+0.4386179;8 middle= @(z) −0.041572∗z.^6+0.43222∗z.^5−1.5111∗z.^4+1.7822∗z

.^3−0.036178∗z.^2+0.094166∗z+0.33886014;9 outlet= @(z) −0.012984∗z.^6+0.13411∗z.^5−0.51332∗z.^4+0.78012∗z

.^3−0.28829∗z.^2+0.20122∗z+0.5568055;10 uniform=@(z) 1+0∗z;11 pow_distrib=uniform;1213 %input variables14 r=bc.D/2; %pipe radius [m]15 length=bc.L; % [m]16 pressure=bc.POUTL; %[N/m2]17 totalflow=bc.FLOWFUE; %total mass flow rate [kg/s]18 total_power=bc.ACTPOW; %total power [J/s]19 segments=bc.CZMESH; %power segments20 distrib=bc.PINPOW; %power distribution2122 N=1000; %number of elements2324 % Properties25 g=9.81; %gravitational acceleration26 rho_liq=XSteam(’rhoL_p’,pressure/1E5); %density of liquid27 rho_vap=XSteam(’rhoV_p’,pressure/1E5); %density of vapor28 t_sat=XSteam(’tsat_p’,pressure/1E5)−0.0001; %saturated temperature29 temp=t_sat−10; % 10K subcooling30 mu_liq=XSteam(’my_pT’,pressure/1E5,t_sat); %saturated liquid viscocity31 hin=XSteam(’h_pT’,pressure/1E5,temp)∗1E3; %enthalpy at inlet [J/kg]

28

32 hl=XSteam(’hL_p’,pressure/1E5)∗1E3; %saturated liquid enthalpy [J/kg]33 hv=XSteam(’hV_p’,pressure/1E5)∗1E3; %saturated vapor enthalpy [J/kg]34 sigma=XSteam(’st_p’,pressure/1E5); %surface tension [N/m]35 quality=@(h) (h−hl)./(hv−hl);36 H=hv−hl; %latent heat of vaporization3738 x_outlet=0.43; %steam quality at outlet39 q=get_heat_flux(pressure,temp,totalflow,x_outlet,r,length); %function to

determine heat flux4041 % Enthalpy and quality distribution42 z=linspace(0,length,N);43 PINPOW=distrib./sum(distrib); % Normalization44 h=zeros(1,N);4546 for i=1:100047 h(i)=hin+2∗pi∗r∗q∗integral(pow_distrib,0,z(i))/totalflow;48 end49 x=quality(h); %steam quality as function of position5051 % Onset of annular flow52 vap_factor=sqrt(rho_vap/(g∗2∗r∗(rho_liq−rho_vap))); %factor multiplied

with J to gain dimensionless J_vapstar53 liq_factor=sqrt(rho_liq/(g∗2∗r∗(rho_liq−rho_vap))); %factor multiplied with

J to gain dimensionless J_liqstar54 x_onset_from_Wallis=(0.4∗pi∗r∗r/totalflow+0.6∗liq_factor/rho_liq)/(

vap_factor/rho_vap+0.6∗liq_factor/rho_liq);55 z_onset_from_Wallis=interp1(x,z,x_onset_from_Wallis); % [m]5657 %1D model58 step_size=length/(N−1);59 d_frac_answer=droplet_fraction(x_onset_from_Wallis,mu_liq,r,

totalflow,rho_liq,rho_vap,sigma); %call droplet fraction function60 Wf_onset=(1−x_onset_from_Wallis)∗(1−d_frac_answer)∗totalflow; %

film flow rate at onset [kg/s]61 Wd_onset=(1−x_onset_from_Wallis)∗d_frac_answer∗totalflow;6263 % check where onset occurs64 k=0;65 onset_value=0;66 while onset_value<z_onset_from_Wallis67 onset_value=onset_value+step_size;68 k=k+1; %onset occurs at k−th element in z69 end7071 %create vectors that goes from onset to end72 dep_okawa=zeros(1,N−k);73 ent_okawa=zeros(1,N−k);74 Wg=zeros(1,N−k+1);75 Wf=zeros(1,N−k+1);

29

76 Wd=zeros(1,N−k);77 film_thickness=zeros(1,N−k+1);78 steamvel=zeros(1,N−k+1);79 film_area=zeros(1,N−k);80 film_velocity=zeros(1,N−k);8182 %entrainment and deposition are first set to zero83 Ment=0;84 Mdep=0;8586 %starting values for vapor and film flow rate87 Wg(1)=x_onset_from_Wallis∗totalflow;88 Wf(1)=Wf_onset;89 film_thickness(1)=r/2; %guess starting value for film thickness90 for i=1:N−k %start loop from onset and onwards91 Wg(i+1)=Wg(1)+(q/H)∗integral(pow_distrib,z(k),z(k+i))∗pi∗2∗r; %vapor

flow rate [kg/s]92 steamvel(i+1)=Wg(i+1)/(rho_vap∗pi∗r∗r); %superfiscial steam velocity [m/s]9394 %call film thickness function95 film_thickness(i+1)=get_film_thickness(Wf(i),film_thickness(i),mu_liq,r,

Wg(i),rho_liq,rho_vap); % [m]96 film_area(i)=pi∗(2∗r∗film_thickness(i+1)−film_thickness(i+1)∗

film_thickness(i+1)); % [m2]97 film_velocity(i)=Wf(i)/(rho_liq∗film_area(i)); % [m/s]9899 Wd(i)=totalflow−Wg(i)−Wf(i); %droplet mass flow [kg/s]

100 ent_okawa(i)=entrainment(film_thickness(i+1),Wg(i),rho_liq,rho_vap,sigma,r); %call entrainment function [kg/m2s]

101 dep_okawa(i)=deposition(Wg(i),Wd(i),rho_liq,rho_vap,sigma,r); %calldeposition function [kg/m2s]

102 Mdep=Mdep+dep_okawa(i)∗step_size∗pi∗2∗r; %add and multiply over area103 Ment=Ment+ent_okawa(i)∗step_size∗pi∗2∗r; %add and multiply over area104105 Wf(i+1)=Wf(1)−Ment+Mdep−Wg(i+1)+Wg(1); %calculate new film flow rate

from mass balance [kg/s]106 if Wf(i+1)<0107 Wf(i+1)=0; %since film flow rate cannot be negative108 end109 end %loop from onset to end of pipe110111 % DETERMINE INITIAL CONDITIONS112 % liquid phase in Main model injected at z=2.59113 % vapor phase in Main model injected at z=2.48114 z_Main_liq=round(interp1(z,(1:N),2.59)); %obtain element in z where z=2.59115 z_Main_vap=round(interp1(z,(1:N),2.48)); %obtain element in z where z=2.48116117 % initial conditions for Main model118 Main_first_inlet_steamvel=steamvel(z_Main_vap−k+1); %vapor velocity

at first Main inlet (z=2.48)

30

119 Main_second_inlet_steamvel=steamvel(z_Main_liq−k+1); %vaporvelocity at second Main inlet (z=2.59)

120 Main_inlet_dropletmassflux=Wd(z_Main_liq−k)/(pi∗r∗r); %droplet massflow [kg/m2s]

121122 % initial film mass flux boundary condition123 injection_wall_area=0.0042847384; % area of injection wall obtained from

Fluent124 %injection_wall_area=0.020812442; % area of of injection wall for "longpipe"−

geometry125 filmBC=Wf(z_Main_liq−k)/injection_wall_area; %film mass flux [kg/m2s]126127 %plot section of pipe from onset to end128 z_annular=(z(k+1):step_size:z(N));129 figure(2)130 plot(z_annular,dep_okawa,’b’,z_annular,ent_okawa,’r’)131 title(’Deposition and entrainment’)132 ylabel(’Deposition and entrainment [kg/m2s]’)133 xlabel(’z [m]’)134 legend(’Deposition’,’Entrainment’)135 grid on136137 %creating vectors for final section of pipe (Main model)138 dep_main=zeros(1,N−z_Main_liq+1);139 ent_main=zeros(1,N−z_Main_liq+1);140 filmthickness_main=zeros(1,N−z_Main_liq+1);141 steamvel_main=zeros(1,N−z_Main_liq+1);142 filmvel_main=zeros(1,N−z_Main_liq+1);143 film_area_main=zeros(1,N−z_Main_liq+1);144 Wf_main=zeros(1,N−z_Main_liq+1);145 Wg_main=zeros(1,N−z_Main_liq+1);146 Wd_main=zeros(1,N−z_Main_liq+1);147 z_main=(z(z_Main_liq):step_size:z(N));148149 %calculating values for final section (last 1.06 m)150 for i=z_Main_liq:N151 dep_main(i−z_Main_liq+1)=dep_okawa(i−k);152 ent_main(i−z_Main_liq+1)=ent_okawa(i−k);153 filmthickness_main(i−z_Main_liq+1)=film_thickness(i−k);154 steamvel_main(i−z_Main_liq+1)=steamvel(i−k+1);155 filmvel_main(i−z_Main_liq+1)=film_velocity(i−k);156 film_area_main(i−z_Main_liq+1)=pi∗(2∗r∗film_thickness(i−k)−

film_thickness(i−k)∗film_thickness(i−k));157 Wf_main(i−z_Main_liq+1)=rho_liq∗filmvel_main(i−z_Main_liq+1)∗

film_area_main(i−z_Main_liq+1);158 Wg_main(i−z_Main_liq+1)=Wg(i−k);159 Wd_main(i−z_Main_liq+1)=Wd(i−k);160 end161162 %plot final section

31

163 figure(3)164 plot(z_main,dep_main,’b’,z_main,ent_main,’r’)165 title(’Deposition and entrainment’)166 ylabel(’Deposition and entrainment [kg/m2s]’)167 xlabel(’z [m]’)168 legend(’Deposition’,’Entrainment’)169 grid on170171 Wf(end)=[];172 Wg(end)=[];173174 figure(4)175 plot(z_annular,Wf,’r’,z_annular,Wg,’k’,z_annular,Wd,’b’)176 title(’Film flow rates’)177 xlabel(’z [m]’)178 ylabel(’Mass flow rates [kg/s]’)179 legend(’Liquid film’,’Vapor’,’Droplets’)180 grid on

1 function [ heat_flux ] = get_heat_flux( pressure,temp,totalflow,x_outlet,r,length )

2 %Input: pressure [Pa], temperature [K], massflow [kg/s],3 %steam quality at outlet [dim−less (%)]4 %Output: heat flux [W/m2]56 hin=XSteam(’h_pT’,pressure/1E5,temp)∗1E3; %enthalpy at inlet [J/kg]7 hf=XSteam(’hL_p’,pressure/1E5)∗1E3; %saturated liquid enthalpy [J/kg]8 hg=XSteam(’hV_p’,pressure/1E5)∗1E3; %saturated vapor enthalpy [J/kg]9 heat_flux=(x_outlet∗(hg−hf)+hf−hin)∗totalflow/(2∗pi∗r∗length);

1011 end

1 function [ d_frac_answer] =droplet_fraction( x_onset_from_Wallis,mu_liq,r,totalflow,rho_liq,rho_vap,sigma )

2 %Output: droplet/film fraction [dim−less]3 %Input: steam quality at onset [dim−less], viscocity at onset [Pa s]45 film_thickness=r/2; %starting value for film thickness (r/2)6 d_step_size=0.001;7 d_frac=d_step_size; %starting value for droplet/film fraction8 dep=zeros(1,1/d_step_size);9 ent=zeros(1,1/d_step_size);

10 Wg=x_onset_from_Wallis∗totalflow; %steam flow rate [kg/s]11 for i=1:1/d_step_size %loop to find droplet fraction12 Wf=(1−x_onset_from_Wallis)∗(1−d_frac)∗totalflow; %film flow rate [kg/s]13 Wd=(1−x_onset_from_Wallis)∗d_frac∗totalflow; %droplet flow rate [kg/s]14 film_thickness=get_film_thickness( Wf,film_thickness,mu_liq,r,Wg,

rho_liq,rho_vap); %call film thickness function15

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16 dep(i)=deposition(Wg,Wd,rho_liq,rho_vap,sigma,r); %call depositionfunction

17 ent(i)=entrainment(film_thickness,Wg,rho_liq,rho_vap,sigma,r); %callentrainment function

18 d_frac=d_frac+d_step_size; %slowly increase droplet/film fraction19 end20 figure; %plot entrainment and deposition as function of droplet/film fraction21 fraction=linspace(d_step_size,d_frac−d_step_size,i);22 plot(fraction,dep,’b’,fraction,ent,’r’)23 legend(’Deposition’,’Entrainment’)24 xlabel(’Droplet/Film fraction’)25 ylabel(’Entrainment and deposition rate [kg/sm2]’)26 grid on27 d_frac_answer=interp1(abs(dep−ent),fraction,min(abs(dep−ent))); %check

where the curves cross28 end

1 function [ film_thickness ] = get_film_thickness( Wf,film_thickness,mu_liq,r,Wg,rho_liq,rho_vap)

2 %Output: film thickness [m]3 %Input: Film flow rate [kg/s], old film thickness [m], viscocity [Pa s],4 %vapor flow rate [kg/s]56 delta=0;7 Re_film=Wf∗4/(2∗r∗pi∗mu_liq); %Reynolds number for film8 if Re_film<=09 film_thickness=0;

10 return11 end12 fw=max(16/Re_film,0.005); %wall friction factor13 while abs(1−delta)>0.00001 %loop to obtain film thickness14 fi=0.005∗(1+150∗film_thickness/r); %interfacial friction factor15 film_thickness2=sqrt(fw∗rho_liq/(fi∗rho_vap))∗(rho_vap∗Wf/(rho_liq∗

Wg))∗r/2; %updated film thickness16 delta=film_thickness2/film_thickness;17 film_thickness=film_thickness2; %repeat until convergence18 end19 end

1 function [ ent ] = entrainment( film_thickness,Wg,rho_liq,rho_vap,sigma,r)

2 % Output: entrainment rate [kg/m2s]3 % Input: film thickness [m], vapor mass flow Wg [kg/s], liquid density [kg/m3]4 % vapor density [kg/m3], surface tension sigma [N/m2]5 ke=0.000479; %mass transfer coefficient [m/s] (constant)6 en=0.111; %exponent (constant)78 fi=0.005∗(1+150∗film_thickness/r); %interfacial friction factor9 Jg=Wg/(rho_vap∗pi∗r∗r); %superfiscial vapor velocity

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10 ent=ke∗rho_liq∗fi∗rho_vap∗Jg∗Jg∗film_thickness∗power((rho_liq/rho_vap),en)/sigma;

11 end

1 function [ dep ] = deposition( Wg,Wd,rho_liq,rho_vap,sigma,r )2 % Output: Deposition rate [kg/m2s]3 % Input: vapor flow rate Wg [kg/s], droplet flow rate Wd [kg/s], liquid density [kg/

m3]4 % vapor density [kg/m3], surface tension sigma [N/m], pipe radius r [m]56 C=Wd/(Wd/rho_liq+Wg/rho_vap); %concentration of droplets [kg/m3]7 dep=0.0632∗sqrt(C∗sigma/(2∗r));%deposition rate [kg/m2s]8 end

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9 Appendix BDuring this work. User Defined Functions were frequently used when performingsimulations in ANSYS Fluent. The example shown below is for the standard casewith G = 1250kg/m2s and steam quality at outlet x = 51% and a transversevelocity for particles at entrainment 60% of original calculated value.

12 #include "udf.h"3 #include "sg_film.h"45 real hvap = 1505132;6 real heat_flux= 983491;7 real n = 0.111;8 real ke = 0.000479;9 real rhovap = 36.525;

10 real rholiq = 739.72;11 real r = 0.007;12 real muliq = 0.000091249;13 real Ref_crit = 320;14 real sigma = 0.017633;15 real p_diam = 0.0012;16 real umean = 11.4915;17 real z_in= 2.8593;181920 DEFINE_PROFILE(film_mass_source,tf,i)21 {22 real v_trans, ufree, ff, utau, film_thickness, film_vel, vap_vel, Jf, Jg, fi,

Ref, area_face, area[ND_ND], pos[ND_ND], r1, x_inj, y_inj, z_inj,evap_mass_source, evap_mom_source, ent_mass_source;

23 face_t f;24 cell_t c;25 Thread ∗tc;2627 begin_f_loop(f,tf)28 {2930 film_thickness = F_EFILM_HEIGHT(f,tf);31 film_vel = F_EFILM_W(f,tf);32 c = F_C0(f,tf);33 tc = THREAD_T0(tf);34 F_CENTROID(pos,f,tf);35 vap_vel = umean+2∗heat_flux∗(pos[2]−z_in)/(hvap∗rhovap∗r); Jf

= 4∗film_vel∗film_thickness/(2∗r);36 Jg = vap_vel;37 fi = 0.005∗(1+300∗(film_thickness/(2∗r)));38 Ref = (rholiq∗Jf∗2∗r)/muliq;39 if (Ref>Ref_crit)40 ent_mass_source = ((ke∗rholiq∗fi∗rhovap∗Jg∗Jg∗

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film_thickness)/sigma)∗pow (rholiq/rhovap,n);41 else42 ent_mass_source = 0.0;434445 evap_mass_source = heat_flux/hvap;4647 evap_mom_source = evap_mass_source∗film_vel;4849 F_AREA(area,f,tf);50 area_face = NV_MAG(area);51 r1 = r−5∗film_thickness;52 x_inj = r1∗(pos[0]/r);53 y_inj = r1∗(pos[1]/r);54 z_inj = pos[2];5556 ufree = vap_vel∗(8/7);57 ff = 0.045/pow(ufree∗film_thickness/(muliq/rholiq),0.25);58 utau=sqrt(ff∗pow(ufree,2.)/2.0);59 v_trans=11.1∗utau∗sqrt(rhovap/rholiq);6061 C_UDMI(c,tc,0) = evap_mass_source;62 C_UDMI(c,tc,1) = evap_mass_source∗area_face;6364 C_UDMI(c,tc,2) = evap_mom_source;65 C_UDMI(c,tc,3) = evap_mom_source∗area_face;6667 C_UDMI(c,tc,4) = ent_mass_source;68 C_UDMI(c,tc,5) = ent_mass_source∗area_face;6970 C_UDMI(c,tc,6) = x_inj;71 C_UDMI(c,tc,7) = y_inj;72 C_UDMI(c,tc,8) = z_inj;7374 C_UDMI(c,tc,9)=v_trans;75 C_UDMI(c,tc,10)=r1;7677 C_UDMI(c,tc,12) = F_EFILM_DPM_MASS_SRC(f,tf)/

area_face;78 C_UDMI(c,tc,13) = F_EFILM_WS(f,tf);7980 F_PROFILE(f,tf,i) = −(ent_mass_source+evap_mass_source);8182 }end_f_loop(f,tf)83 }848586 DEFINE_DPM_INJECTION_INIT(entrainment_inj,I)87 {88 Particle ∗p;

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89 cell_t c;90 Thread ∗tc;9192 loop(p,I−>p_init)93 {9495 c = P_CELL(p);96 tc = P_CELL_THREAD(p);97 P_POS(p)[0]=C_UDMI(c,tc,6);98 P_POS(p)[1]=C_UDMI(c,tc,7);99 P_POS(p)[2]=C_UDMI(c,tc,8);

100 P_FLOW_RATE(p)= C_UDMI(c,tc,5);101 P_DIAM(p)=p_diam;102 P_VEL(p)[0]=0.6∗C_UDMI(c,tc,9)∗(−C_UDMI(c,tc,6))/C_UDMI(c,tc

,10);103 P_VEL(p)[1]=0.6∗C_UDMI(c,tc,9)∗(−C_UDMI(c,tc,7))/C_UDMI(c,tc

,10);104 P_VEL(p)[2]=C_W(c,tc);105106 C_UDMI(c,tc,11)=C_W(c,tc);107 }108 }109110 DEFINE_SOURCE(vapor_mass_source,c,tc,dS,eqn)111 {112113 real vol_cell,evap_mass_source;114 vol_cell = C_VOLUME(c,tc);115 evap_mass_source = C_UDMI(c,tc,1)/vol_cell;116 return evap_mass_source;117 }118119120 DEFINE_SOURCE(vapor_mom_source,c,tc,dS,eqn)121 {122 real vol_cell,evap_mom_source;123124 vol_cell = C_VOLUME(c,tc);125 evap_mom_source = C_UDMI(c,tc,3)/vol_cell;126 return evap_mom_source;127 }128129 DEFINE_PROFILE(z_film_mom_source,tf,i)130 {131132 face_t f;133 cell_t c;134 Thread ∗tc;135 real drop_vel, ent_flux;136

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137 begin_f_loop(f,tf)138 {139140 c = F_C0(f,tf);141 tc = THREAD_T0(tf);142143 drop_vel = C_UDMI(c,tc,11);144 ent_flux = C_UDMI(c,tc,4);145146 F_PROFILE(f,tf,i)= −(C_UDMI(c,tc,2)+ent_flux∗drop_vel);147 }end_f_loop(f,tf)148 }149150 DEFINE_PROFILE(y_film_mom_source,tf,i)151 {152153 face_t f;154 cell_t c;155 Thread ∗tc;156 real drop_vel, ent_flux;157158 begin_f_loop(f,tf)159 {160161 c = F_C0(f,tf);162 tc = THREAD_T0(tf);163164 drop_vel = 0;165 ent_flux = C_UDMI(c,tc,4);166167 F_PROFILE(f,tf,i)= −ent_flux∗drop_vel;168 }end_f_loop(f,tf)169 }170171 DEFINE_PROFILE(x_film_mom_source,tf,i)172 {173174 face_t f;175 cell_t c;176 Thread ∗tc;177 real drop_vel, ent_flux;178179 begin_f_loop(f,tf)180 {181182 c = F_C0(f,tf);183 tc = THREAD_T0(tf);184185 drop_vel = 0;186 ent_flux = C_UDMI(c,tc,4);

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187188 F_PROFILE(f,tf,i)= −ent_flux∗drop_vel;189 }end_f_loop(f,tf)190 }

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References[1] Carl Adamsson and Henryk Anglart. Film flow measurements for high-

pressure diabatic annular flow in tubes with various axial power distribu-tions. Nuclear Engineering and Design, 236:2485–2493, 2006.

[2] P. Andreussi and B.J. Azzopardi. On the entrainment of drops by the gasin two-phase annular flow. Chemical Engineering Science, 39(9):1426–1428,1984.

[3] Inc. 2017 ANSYS. Ansys fluent theory guide 18.0.

[4] Maria Camacho. CFD modeling of annular flow to predict behavior of theliquid film. Master’s thesis, KTH, Stockholm, Sweden, 2016.

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[9] Hideaki UTSUNO and Fumito KAMINAGA. Prediction of Liquid FilmDryout in Two-Phase Annular-Mist Flow in a Uniformly Heated NarrowTube Development of Analytical Method under BWR Conditions. Journalof Nuclear Science and Technology, 35(9):643–653, 1998.

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CFD Annular Flow Modelling Based on a Three-Field Approach

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