رسالة دكتوراة بعنوان CFD Modeling of Two-Phase Boiling Flows in the Slug Flow Regime with an Interface Capturing Technique

رسالة دكتوراة بعنوان CFD Modeling of Two-Phase Boiling Flows in the Slug Flow Regime with an Interface Capturing Technique
اسم المؤلف
MIRCO MAGNINI
التاريخ
3 نوفمبر 2020
المشاهدات
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رسالة دكتوراة بعنوان
CFD Modeling of Two-Phase Boiling Flows in the Slug Flow Regime with an Interface Capturing Technique
Alma Mater Studiorum – Università di Bologna
DOTTORATO DI RICERCA IN
Ingegneria Energetica, Nucleare e del
Controllo Ambientale
Ciclo XXIV
Settore Concorsuale: 09/C2 – Fisica Tecnica
Settore Scientifico-disciplinare: ING-IND/10 – Fisica Tecnica Industriale
Presentata da: MIRCO MAGNINI
Coordinatore Dottorato: Relatori:
Prof. Antonio Barletta Dr. Beatrice Pulvirenti
Prof. John R. Thome
Contents
Acknowledgments iii
Abstract v
Sommario vii
Contents ix
List of Tables xiii
List of Figures xv
Nomenclature xxi
Introduction xxvii
1 Mathematical formulation of two-phase flows 1
1.1 Formulation without surface tracking 2
1.2 Formulation with surface tracking 4
1.2.1 Two-fluid formulation 5
1.2.2 Single fluid formulation . 8
1.3 The Front Tracking algorithm 11
1.4 The Level-Set method 13
1.5 The Volume Of Fluid method 15
1.6 Hybrid methods . 19
1.7 Surface tension force modeling . 20
ixx CONTENTS
2 Elongated bubbles flow: a review 25
2.1 Vertical circular channels 26
2.1.1 Experiments and analytical models . 27
2.1.2 Numerical simulations 30
2.2 Horizontal circular channels . 33
2.2.1 Experiments and analytical models . 34
2.2.2 Numerical simulations 49
2.3 Concluding remarks: open issues on flow boiling in microscale 56
3 Modeling of interfacial effects 57
3.1 The interface reconstruction algorithm . 58
3.1.1 The Height Function algorithm . 62
3.2 The evaporation model 69
3.2.1 The interface temperature condition 69
3.2.2 The numerical model . 74
4 Ansys Fluent solver and the implementation of the UDF 79
4.1 The flow equation set 80
4.2 Fluent discretization procedure . 81
4.2.1 Temporal discretization . 82
4.2.2 Spatial discretization . 83
4.2.3 Reconstruction of the cell centered gradients . 84
4.2.4 The final algebraic equation . 86
4.3 Discretization of the volume fraction equation . 87
4.4 Pressure-velocity coupling: the PISO algorithm 92
4.4.1 Interpolation of cell-centered pressures on cell faces 95
4.5 The energy equation . 96
4.6 The additional scalar equation for the evaporation rate 96
4.7 Boundary conditions . 97
4.8 User-Defined Functions . 99
4.8.1 Initialization of the volume fraction field 101
4.8.2 Implementation of the surface tension force 101
4.8.3 Implementation of the evaporation model . 102
4.9 Fluent solution procedure 103CONTENTS xi
4.10 Concluding remarks: flow solver set-up . 107
5 Validation of the numerical framework 109
5.1 Reproduction of a circular interface . 109
5.2 Simulation of an inviscid static droplet . 112
5.3 Isothermal bubble rising in stagnant liquid . 117
5.3.1 Two-dimensional inviscid rising bubble . 117
5.3.2 Axisymmetrical bubble rising in viscous liquid 119
5.4 Vapor bubble growing in superheated liquid 123
5.4.1 Discrete domain and initial conditions . 124
5.4.2 Initial thermal boundary layer placement . 125
5.4.3 Working fluids properties 126
5.4.4 Setting of diffusion parameter 127
5.4.5 Results 128
5.5 Concluding remarks . 131
6 Results on elongated bubbles motion in adiabatic condition 133
6.1 Taylor bubbles rising in vertical circular channels . 133
6.1.1 Simulation sensitivity analysis 134
6.1.2 Comparison with PIV analysis . 137
6.1.3 Numerical simulations of Taylor bubbles: results . 140
6.2 Elongated bubbles in horizontal circular channels . 144
6.3 Concluding remarks: limits of the computations 153
7 Results on elongated bubbles motion with evaporation 155
7.1 Grid convergence analysis 156
7.2 Flow boundary conditions 165
7.3 Flow and temperature field . 172
7.3.1 Comparison of bubble nose position with a theoretical model 185
7.4 Heat transfer with multiple bubbles 188
7.4.1 Simulation conditions 188
7.4.2 Bubbles dynamics 190
7.4.3 Heat transfer performance 193
Conclusions 205A Numerically induced capillary waves in the simulation of multiphase
flows 211
A.1 Stability analysis of a static droplet . 212
A.2 Numerical origin of the capillary wave . 215
A.3 Numerical simulations of the static droplet 216
A.3.1 Oscillation time period 217
A.3.2 Droplet profile evolution . 218
A.3.3 Velocity fields . 220
Bibliography 221
List of Scientific Publications 235
Curriculum Vitae 237List of Tables
3.1 Relative magnitude of microscale effects on the interface temperature
deviation from the saturation condition . 72
4.1 Comparison of the Green-Gauss cell based and node based schemes
for the reconstruction of cell centered gradients . 86
4.2 Comparison of the Fluent body-force-weighted and PRESTO formulations to compute face-centered pressures 95
5.1 Comparison of experimental and numerical terminal shape and Reynolds
number for gas bubbles rising in stagnant viscous liquids . 120
5.2 Properties of the working fluids chosen for the simulation of a vapor
bubble growing in superheated liquid 126
6.1 Parameters varied in the rising Taylor bubble sensitivity analysis . 135
6.2 Comparison of numerical results with experimental correlations for
flow within horizontal channels 148
7.1 Properties of R113 liquid and vapor at saturation conditions for Tsat =
50 oC 157
7.2 Properties of R245fa liquid and vapor at saturation conditions for
Tsat = 31 oC 189
7.3 Heat transfer coefficients at the axial locations under analysis . 196
7.4 Comparison of the heat transfer performance obtained with the simulation with the values predicted by correlations 199
A.1 Summary of the parameters varied in the numerical simulations . 217
A.2 Comparison of analytical and numerical periods of the oscillations 219
xiiiList of Figures
1.1 Example of computational grids for BFM and ALE method 7
1.2 Example of moving front and fixed grid for the Front Tracking method. 11
1.3 Example of the level-set field across an interface 13
1.4 Example of the volume fraction field across an interface 16
1.5 Different VOF-based interface reconstruction methods . 18
1.6 Spurious velocity field across a circular interface 22
2.1 White and Beardmore [1] flow pattern map for Taylor bubbles rising
in stagnant liquid . 28
2.2 Revellin et al. [2] elongated bubble images for R134a at 30 oC within
a 2 mm, 0:8 mm and 0:5 mm diameter channel . 34
2.3 Han and Shikazono transition map [3] for the influence of gravitational
effects on the slug flow within horizontal microchannels 35
2.4 Han and Shikazono [3] experimental measured liquid film thickness
for slug flow within horizontal microchannels 39
2.5 Microchannel slug flow snapshot for R245fa, horizontal 0:5 mm circular channel, G = 517 kg/m2s, x = 0:047, Tsat = 34:4 oC 40
2.6 Flow pattern map of Triplett et al. [4] for air-water flow in a 1:1 mm
horizontal circular channel 41
2.7 Comparison of the predicted bubble velocity with respect to the liquid
mean velocity, for various models . 44
2.8 Scheme of the bubble-liquid slug unit in the Thome et al. three-zones
model [5] 46
2.9 Walsh et al. [6] experimental local Nusselt number for slug flow in the
thermal entrance zone of a horizontal microchannel 47
xvxvi LIST OF FIGURES
3.1 Example of the volume fraction field across a circular arc and Height
Function approximated interface . 61
3.2 Sketch of the continuous height function . 62
3.3 Examples of Height Function algorithm steps . 68
3.4 Scaling effect on the contribution of different physical effects on the
interfacial superheating 73
3.5 Steps of Hardt and Wondra evaporation model [7] for the derivation
of the mass source term 77
4.1 Example of a computational control volume . 83
4.2 Geometrical reconstruction of the volume fraction equation convective
term by an Eulerian split advection technique . 90
4.3 Fluent pressure-based segregated solution procedure for VOF-treated
two-phase flows 105
5.1 HF and Youngs computed norm vector error norm convergence rate
in the reproduction of a circular interface 111
5.2 HF and Youngs computed curvature error norm convergence rate in
the reproduction of a circular interface . 111
5.3 Velocity error norm in the simulation of an inviscid static droplet for
HF and Youngs methods . 114
5.4 Velocity field arising in the simulation of an inviscid static droplet
when computing the interface curvature by the HF algorithm . 114
5.5 Interface pressure jump error norm in the simulation of an inviscid
static droplet for HF and Youngs methods . 115
5.6 Pressure profiles across a droplet in the simulation of an inviscid static
droplet for HF and Youngs algorithms 116
5.7 Terminal bubble shapes for the simulations of an inviscid rising bubble
in stagnant liquid . 118
5.8 Velocity vectors around a gas bubble rising in viscous stagnant liquid
at low and high Morton numbers . 122
5.9 Initial condition for the simulation of a vapor bubble growing in superheated liquid 124LIST OF FIGURES xvii
5.10 Initial dimensionless temperature profile at the bubble interface on
the liquid side . 125
5.11 Numerical smoothing of the evaporation rate at the interface . 127
5.12 Temperature and vapor volume fraction profiles across the interface
at different time instants . 128
5.13 Velocity vectors and bubble interface positions for water bubble simulation at various time instants 129
5.14 Vapor bubble radius over time for analytical and numerical solutions. 130
6.1 Bubble initial configuration for the simulation of Taylor bubbles rising
in vertical channels 134
6.2 Sensitivity analysis results 137
6.3 Liquid flow field around a Taylor bubble rising in stagnant liquid . 138
6.4 Velocity profiles of the liquid above a rising Taylor bubble 139
6.5 Velocity profiles of the liquid within the liquid film and below a rising
Taylor bubble . 140
6.6 Terminal shape of the Taylor bubbles for the numerical simulations. 142
6.7 Location of the rising Taylor bubbles simulation runs within the White
and Beardmore [1] flow pattern map . 143
6.8 Bubble terminal shapes for flow within horizontal channels 146
6.9 Static pressure profiles along the channel axis for flow within horizontal channels . 147
6.10 Pressure field and velocity vectors across the bubble for Ca= 0:0125
and Re= 625 150
6.11 Profiles of the dimensionless axial velocity of the liquid along the radial
direction at various axial locations in the wake and in the liquid film
for Ca= 0:0125 and Re= 625 . 151
6.12 Streamlines of the defect flow field in the wake and the film region for
Ca= 0:0125 and Re= 625 . 152
7.1 Initial configuration for the simulations involved in the grid convergence analysis . 158
7.2 Initial wall temperature and heat transfer coefficient 158
7.3 Velocity of the bubble nose 159xviii LIST OF FIGURES
7.4 Bubble profiles at t = 5:5 ms . 159
7.5 Bubble volume growth rate 160
7.6 Bubble evolution at various time instants 161
7.7 Heat transfer coefficients at t = 7:5; 8:5; 9:5 ms . 163
7.8 Heat transfer coefficients at t = 10:5; 11:5; 12:5 ms . 164
7.9 Bubble evolution during evaporation, from t = 4:5 ms to 12:5 ms at
time intervals of 1 ms . 166
7.10 Bubble volume growth rate and velocity of rear, nose and center of
gravity . 167
7.11 Bubble profiles at various time instants obtained with different boundary conditions . 169
7.12 Bubble evolution during evaporation, from t = 4:5 ms to 14:5 ms at
time intervals of 1 ms, q = 20 kW/m2 171
7.13 Velocity of the bubble rear and nose, q = 20 kW/m2 172
7.14 Average liquid axial and radial velocity and contours of the velocity
field . 174
7.15 Wall temperature, heat transfer coefficient and temperature contours
in the wake . 175
7.16 Wall temperature, heat transfer coefficient and temperature contours
in the region occupied by the bubble . 176
7.17 Local enhancement on the heat transfer induced by the two-phase flow.177
7.18 Temperature and axial velocity profiles at z=D = 10; 11; 12:4 . 178
7.19 Temperature and axial velocity profiles at z=D = 12:65; 12:83; 13:5 179
7.20 Temperature and axial velocity profiles at z=D = 14; 14:5; 19 . 180
7.21 Fluid flow, temperature field and heat transfer in the wavy region of
the film . 182
7.22 Comparison of the local bulk heat transfer coefficient with the heat
conduction based heat transfer coefficient 183
7.23 Comparison of bubble nose position and volume with Consolini and
Thome model [8] 186
7.24 Heat flux absorbed by the bubble evaporation . 187
7.25 Evolution of two bubbles flowing in sequence during evaporation . 191
7.26 Leading and trailing bubbles position and velocity during evaporation. 1927.27 Profiles of the bubble before and at the end of the heated region . 193
7.28 Heat transfer coefficient at various axial locations . 195
7.29 Time-averaged heat transfer coefficient . 197
7.30 Comparison of the simulation heat transfer coefficient with the model. 202
A.1 Initial non-dimensional droplet radius distribution . 215
A.2 Numerical dimensionless velocity norm with respect to the non-dimensional
time . 218
A.3 Non-dimensional droplet radius evolution in time 220
Nomenclature
Roman Letters
Af area of the computational cell face [m2]
Bo = ρgD
σ Bond number [-]
b generic fluid property
Ca = µU
σ Capillary number [-]
Co = g(ρl−σρg)D2 1=2 Confinement number [-]
Co = ∆t
V= P
Nf
f uf ·nf Af
Courant number [-]
c
p constant pressure specific heat [m2 s−2 K−1]
D diameter [m]
diffusion constant [m2]
E interfacial energy transfer [kg m−1 s−3]
mass average energy [m2 s−2]
Eo = ρgD2
σ E¨otv¨os number [-]
e specific internal energy [m2 s−2]
Fr = pUgD Froude number [-]
G mass flux [kg m−2 s−1]
g gravity vector [m s−2]
H level-set smoothed Heaviside function [-]
height function [m]
h heat transfer coefficient [kg s−3 K−1]
grid spacing [m]
specific enthalpy [m2 s−2]
xxixxii NOMENCLATURE
hlv latent heat of vaporization [m2 s−2]
I indicator function [-]
I identity tensor [-]
L length [m]
Ls liquid slug length [m]
M molecular weight [kg mol−1]
M interfacial momentum transfer vector [kg m−2 s−2]
Mo = gµ4
ρσ3 Morton number [-]
m bubble relative drift velocity [-]
m_ interphase mass flux [kg m−2 s−1]
N,Nl,Nv normalization factors [-]
Nu = hD
λ Nusselt number [-]
Nf = ρg1=2µD3=2 inverse viscosity number [-]
n interface unit norm vector [-]
Pe = ρcpUD
λ Peclet number [-]
Pr = µcp
λ Prandtl number [-]
p pressure [kg m−1 s−2]
q heat flux [kg s−3]
q_ interphase heat flux [kg s−3]
R radius [m]
universal gas constant [kg m2 s−2 mol−1 K−1]
Re = ρUD
µ
Reynolds number [-]
r radial coordinate [m]
SE energy source term [kg m−1 s−3]
Sm momentum source term [kg m−2 s−2]
Sα volume fraction source term [kg m−3 s−1]

mass source term [kg m−3 s−1]
S
’ evaporation rate equation source term [kg m−3 s−1]
T temperature [K]
t time [s]
t interface unit tangent vector [-]
U velocity [m s−1]NOMENCLATURE xxiii
u velocity vector [m s−1]
V volume [m3]
V interface velocity vector [m s−1]
We = ρU2D
σ Weber number [-]
x x coordinate [m]
mass fraction [-]
x position vector [m]
y y coordinate [m]
z axial coordinate [m]
zh axial coordinate relative to the entrance [m]
in the heated region
Greek Letters
α volume fraction [-]
β gas phase volumetric flow rate [-]
Scriven model growth constant [-]
Γ interfacial mass transfer [kg m−3 s−1]
γ thermal diffusivity [m2 s−1]
∆x,∆y horizontal and vertical grid spacing [m]
δ Dirac delta-function
liquid film thickness [m]
δT thickness of the thermal boundary layer [m]
 void fraction [-]
κ interface curvature [m−1]
λ thermal conductivity [kg m s−3 K−1]
µ dynamic viscosity [kg m−1 s−1]
ρ density [kg m−3]
σ surface tension coefficient [kg s−2]
accommodation coefficient [-]xxiv NOMENCLATURE
τ shear stress tensor [kg m−1 s−2]
φ level-set function [m]
kinetic mobility [kg m−2 s−1 K−1]
generic flow variable
’,’0 smeared and original evaporation rate [kg m−3 s−1]
Subscripts
b bubble, bulk
c cell-centroid-centered value
ex exact value
f cell-face-centered value
G center of gravity
g gas
if relative to the interface
l liquid
N bubble nose
n node-centered value
R bubble rear
ref reference conditions
s superficial
sat saturation conditions
sp single phase
tp two-phase
v vapor
w wall
1 relative to system ambient conditionsAcronyms
ALE Arbitrary Lagrangian Eulerian
BFM Boundary Fitted Method
CD Centered finite Difference
CFD Computational Fluid Dynamics
CLSVOF Coupled Level-Set and Volume Of Fluid
CSF Continuum Surface Force Method
CSS Continuum Surface Stress Method
ENO Essentially Non-Oscillatory
FT Front Tracking
GSM Ghost Fluid Method
HF Height Function
LS Level-Set
LVIRA Least-squares Volume of fluid Interface Reconstruction Algorithm
MAC Marker and Cell
MPI Message Passing Interface
MUSCL Monotonic Upstream-centered Scheme for Conservation Laws
PDA Photocromic Dye Activation
PISO Pressure Implicit Splitting of Operators
PIV Particle Image Velocimetry
PLIC Piecewise Linear Interface Calculation
PRESTO PRessure STaggering Option
PROST Parabolic Reconstruction Of Surface Tension
RPI Renssealer Polytechnic Institute
SIMPLE Semi Implicit Method for Pressure-Linked Equations
SIMPLEC SIMPLE Consistent
SLIC Single Line Interface Calculation
SOU Second Order Upwind
UDF User Defined Function
VOF Volume Of Fluid
WENO Weighted Essentially Non-Oscillatory
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