Mathematical Modeling of Convective Heat Transfer:
From Single Phase to Subcooled Boiling Flows
by
Matthew P. Wilcox
A Thesis Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF SCIENCE
Major Subject: MECHANICAL ENGINEERING
Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Thesis Adviser
Rensselaer Polytechnic Institute
Hartford, Connecticut
May 2013
i
© Copyright 2013
By
Matthew P. Wilcox
All Rights Reserved
ii
TABLE OF CONTENTS
LIST OF TABLES ............................................................................................................. v
LIST OF FIGURES ......................................................................................................... vii
LIST OF SYMBOLS ......................................................................................................... x
ABSTRACT .................................................................................................................... xii
1. INTRODUCTION ....................................................................................................... 1
1.1
SUMMARY OF PRIOR WORK ....................................................................... 2
1.2
CONTENT ......................................................................................................... 3
2. HEAT TRANSFER AND FLUID FLOW: THEORY ................................................ 5
2.1
GOVERNING EQUATIONS ............................................................................ 5
2.2
NUMERICAL METHODS................................................................................ 6
2.3
NATURAL CONVECTION .............................................................................. 9
2.4
LAMINAR FLOW ........................................................................................... 11
2.5
TURBULENT FLOW ...................................................................................... 12
2.5.1
2.6
2.7
CALCULATING TURBULENCE PARAMETERS .......................... 14
TWO-PHASE FLOW ...................................................................................... 16
2.6.1
MODELING TWO-PHASE FLOW .................................................... 18
2.6.2
POPULATION BALANCE MODEL.................................................. 19
BOILING HEAT TRANSFER ........................................................................ 20
2.7.1
SUBCOOLED BOILING .................................................................... 22
3. HEAT TRANSFER AND FLUID FLOW: MODELING ......................................... 25
3.1
NATURAL CONVECTION ............................................................................ 25
3.1.1
HORIZONTAL CYLINDER ............................................................... 25
3.1.2
VERTICAL PLATE ............................................................................ 32
3.2
LAMINAR FLOW ........................................................................................... 38
3.3
TURBULENT FLOW ...................................................................................... 42
3.3.1
TURBULENT FLOW WITHOUT HEAT TRANSFER ..................... 42
iii
3.3.2
3.4
3.5
TURBULENT FLOW WITH HEAT TRANSFER ............................. 47
TWO-PHASE FLOW ...................................................................................... 50
3.4.1
GAS MIXING TANK .......................................................................... 50
3.4.2
BUBBLE COLUMN ............................................................................ 56
3.4.3
BUBBLE COLUMN WITH POPULATION BALANCE MODEL ... 61
BOILING HEAT TRANSFER ........................................................................ 65
3.5.1
POOL BOILING .................................................................................. 65
3.5.2
SUBCOOLED FLOW BOILING ........................................................ 71
4. DISUSSION AND CONCLUSIONS ........................................................................ 84
REFERENCES ................................................................................................................ 86
iv
LIST OF TABLES
Table 2.5.1-1: Turbulent Flow Input ............................................................................... 15
Table 2.5.1-2: Calculation of Turbulent Parameters ....................................................... 15
Table 3.1.1-1: Horizontal Cylinder Model Input ............................................................. 26
Table 3.1.1-2: Horizontal Cylinder Model Fluid Density ............................................... 26
Table 3.1.1-3: Mesh Validation for Horizontal Cylinder Model ..................................... 31
Table 3.1.2-1: Vertical Plate Model Input ....................................................................... 33
Table 3.1.2-2: Vertical Plate Model Fluid Density.......................................................... 33
Table 3.1.2-3: Mesh Validation for Vertical Plate Model ............................................... 37
Table 3.2-1: Laminar Flow Model Input ......................................................................... 39
Table 3.2-2: Laminar Flow Model Fluid Density ............................................................ 39
Table 3.2-3: Mesh Validation for Laminar Flow Model ................................................. 41
Table 3.3.1-1: Turbulent Flow Without Heat Transfer Model Input ............................... 43
Table 3.3.2-1: Turbulent Flow With Heat Transfer Model Input .................................... 48
Table 3.3.2-2: Turbulent Flow With Heat Transfer Model Fluid Density ...................... 48
Table 3.3.2-3: Mesh Validation for Turbulent Flow With Heat Transfer Model ............ 50
Table 3.4.1-1: Gas Mixing Tank Model Input ................................................................. 52
Table 3.4.1-2: Mesh Validation for Gas Mixing Tank Model ......................................... 55
Table 3.4.2-1: Bubble Column Model Input ................................................................... 57
Table 3.4.2-2: Mesh Validation for Bubble Column Model ........................................... 61
Table 3.4.3-1: Population Balance Model Input .............................................................. 62
Table 3.4.3-2: Bubble Size Distribution – Surface Tension of 0.072 N/m ..................... 64
Table 3.4.3-3: Bubble Size Distribution – Surface Tension of 0.0072 N/m ................... 65
Table 3.5.1-1: Pool Boiling Model Input......................................................................... 66
Table 3.5.1-2: Pool Boiling Model Fluid Density ........................................................... 67
Table 3.5.1-3: Mesh Validation for Pool Boiling Model................................................. 70
Table 3.5.2-1: Subcooled Flow Boiling Model Input ...................................................... 71
Table 3.5.2-2: Subcooled Flow Boiling Model Fluid Properties..................................... 72
Table 3.5.2-3: Boiling Model Study Case Input .............................................................. 73
Table 3.5.2-4: Boiling Model Study Case Results .......................................................... 76
Table 3.5.2-5: Inlet Condition Study Case Input ............................................................. 78
v
Table 3.5.2-6: Inlet Condition Study Case Results.......................................................... 79
Table 3.5.2-7: Axial Liquid Volume Fraction ................................................................. 79
Table 3.5.2-8: Absolute Impact on Liquid Volume Fraction .......................................... 82
Table 3.5.2-9: Relative Impact on Liquid Volume Fraction ........................................... 82
Table 3.5.2-10: Mesh Validation for Subcooled Flow Boiling Model ............................ 83
vi
LIST OF FIGURES
Figure 2.2-1: Control Volume Schematic for Pressure Correction Equation .................... 7
Figure 2.2-2: Control Volume Schematic for Momentum Equation ................................. 8
Figure 2.2-3: Control Volume Schematic for Energy Equation ........................................ 8
Figure 2.5-1: Transition from Laminar to Turbulent Flow.............................................. 12
Figure 2.6-1: Two-Phase Flow Patterns .......................................................................... 16
Figure 2.6-2: Baker Flow Pattern .................................................................................... 17
Figure 2.7-1: Boiling Heat Transfer Regimes ................................................................. 20
Figure 3.1.1-1: Horizontal Cylinder Schematic ............................................................... 25
Figure 3.1.1-2: Temperature (K) ..................................................................................... 27
Figure 3.1.1-3: Density (kg/m3) ....................................................................................... 27
Figure 3.1.1-4: Velocity Vectors (m/s) ............................................................................ 28
Figure 3.1.1-5: Interference Fringes Around a Heated Horizontal Cylinder ................... 29
Figure 3.1.1-6: Dimensionless Temperature at θ = 30° ................................................... 30
Figure 3.1.1-7: Dimensionless Temperature at θ = 90° ................................................... 30
Figure 3.1.1-8: Dimensionless Temperature at θ = 180° ................................................. 31
Figure 3.1.2-1: Vertical Plate Schematic ......................................................................... 32
Figure 3.1.2-2: Temperature (K) ..................................................................................... 34
Figure 3.1.2-3: Velocity Vectors (m/s) ............................................................................ 34
Figure 3.1.2-4: Interference Fringes Around a Heated Vertical Plate ............................. 35
Figure 3.1.2-5: Dimensionless Temperature for Various Prandtl Numbers .................... 36
Figure 3.1.2-6: Dimensionless Temperature for Various Prandtl Numbers (Fluent) ...... 37
Figure 3.2-1: Laminar Flow Schematic ........................................................................... 38
Figure 3.2-2: Velocity Magnitude ................................................................................... 38
Figure 3.2-3: Radial Velocity (m/s) ................................................................................. 40
Figure 3.2-4: Temperature (K) ........................................................................................ 40
Figure 3.2-5: Wall Shear Stress ....................................................................................... 41
Figure 3.3.1-1: Turbulent Flow Without Heat Transfer Schematic................................. 42
Figure 3.3.1-2: Velocity Magnitude ................................................................................ 42
Figure 3.3.1-3: Wall Shear Stress .................................................................................... 44
Figure 3.3.1-4: Radial Velocity (m/s) ............................................................................. 44
vii
Figure 3.3.1-5: 𝑑𝑢𝑥 ⁄𝑑𝑥 ................................................................................................... 44
Figure 3.3.1-6: Results for a Mass Flow Rate of 0.5 kg/s ............................................... 45
Figure 3.3.1-7: Results for a Mass Flow Rate of 1.5 kg/s ............................................... 45
Figure 3.3.1-8: Turbulent Kinetic Energy (m2/s2) ........................................................... 46
Figure 3.3.1-9: Production of Turbulent Kinetic Energy ................................................ 46
Figure 3.3.2-1: Turbulent Flow With Heat Transfer Schematic ...................................... 47
Figure 3.3.2-2: Temperature (K) ..................................................................................... 47
Figure 3.3.2-3: Radial Velocity (m/s) .............................................................................. 49
Figure 3.3.2-4: Velocity Magnitude ................................................................................ 49
Figure 3.3.2-5: Wall Shear Stress .................................................................................... 49
Figure 3.4.1-1: Gas Mixing Tank Schematic................................................................... 51
Figure 3.4.1-2: Gas Volume Fraction .............................................................................. 53
Figure 3.4.1-3: Gas Volume Fraction at Jet Centerline ................................................... 53
Figure 3.4.1-4: Liquid Velocity Vectors (m/s) ................................................................ 54
Figure 3.4.1-5: Gas Velocity Vectors (m/s)..................................................................... 55
Figure 3.4.2-1: Bubble Column Schematic ..................................................................... 56
Figure 3.4.2-2: Gas Volume Fraction .............................................................................. 58
Figure 3.4.2-3: Liquid Velocity Vectors (m/s) ................................................................ 59
Figure 3.4.2-4: Gas Velocity Vectors (m/s)..................................................................... 60
Figure 3.4.2-5: Gas Volume Fraction (0.10 m/s)............................................................. 60
Figure 3.4.3-1: Gas Volume Fraction with PBM ............................................................ 62
Figure 3.4.3-2: Liquid Velocity Vectors with PBM (m/s)............................................... 63
Figure 3.4.3-3: Gas Velocity Vectors with PBM (m/s) ................................................... 64
Figure 3.5.1-1: Pool Boiling Schematic .......................................................................... 66
Figure 3.5.1-2: Vapor Volume Fraction .......................................................................... 68
Figure 3.5.1-3: Liquid Velocity Vectors (m/s) ................................................................ 69
Figure 3.5.1-4: Vapor Velocity Vectors (m/s) ................................................................. 69
Figure 3.5.1-5: Volume Fraction of Vapor on Heated Surface ....................................... 70
Figure 3.5.2-1: Subcooled Flow Boiling Model Schematic ............................................ 71
Figure 3.5.2-2: Case 1 - Temperature (K) ....................................................................... 73
Figure 3.5.2-3: Case 1 - Liquid Volume Fraction ........................................................... 73
viii
Figure 3.5.2-4: Case 1 - Mass Transfer Rate (kg/m3-s) ................................................... 74
Figure 3.5.2-5: Case 1 - Vapor Generation Rate ............................................................. 75
Figure 3.5.2-6: Liquid Volume Faction for Cases 1-6..................................................... 77
Figure 3.5.2-7: Liquid Volume Faction for Cases 7-12................................................... 80
ix
LIST OF SYMBOLS
A
flow area (m2)
a
cylinder diameter (m)
α
thermal diffusivity (m2/s)
β
coefficient of thermal expansion (K-1)
Cp
specific heat at constant pressure (J/kg-K)
𝜕
partial differential
D/Dt
substantial differential with respect to time
D
pipe diameter (m)
Dh
hydraulic diameter (m)
dbw
bubble departure diameter (m)
ϵ
turbulent dissipation rate (m2/s3)
f
bubble departure frequency (s-1)
g
acceleration due to gravity (m/s2)
g
subscript referring to gas/vapor
h
interfacial heat transfer coefficient (W/m2-K)
hfg
latent heat of vaporization (J/kgmol)
I
turbulent intensity
k
thermal conductivity (W/m-K)
𝑘
turbulent kinetic energy (m2/s2)
l
turbulence length scale (m)
l
subscript referring to liquid
L
length (m)
𝑚̇
mass flow rate (kg/s)
Na
nucleation site density (m-2)
P
perimeter (m)
p
pressure (Pa)
ρ
density (kg/m3)
𝑞⃑
heat flux in vector form (W/m2)
Qw
wall heat flux (W/m2)
r
radial distance in cylindrical coordinates (m)
x
rs
radius of circular pipe (m)
σ
surface tension (N/m)
S
suppression factor
t
time (s)
T
temperature (K)
Twall
wall temperature (K)
𝑇∞
bulk fluid temperature (K)
Tsat
fluid saturation temperature (K)
Tsub
liquid subcooling temperature (K)
θ
contact angle (radians)
𝑢
generalized velocity (m/s)
𝑢𝑥
axial velocity (m/s)
𝑢𝑖
velocity in x-direction (m/s)
𝑢𝑗
velocity in y-direction (m/s)
𝑢̅
time-mean velocity (m/s)
𝑢′
fluctuating component of velocity (m/s)
μ
viscosity (kg/m-s)
⃑⃑
v
average mass velocity in vector form (m/s)
V
mean velocity (m/s)
⃑⃑
∇
del operator
𝜙
scalar quantity
xi
distance in x-direction (m)
xj
distance in y-direction (m)
x
spatial coordinate in a Cartesian or cylindrical system (m)
y
spatial coordinate in a Cartesian system (m)
xi
ABSTRACT
Various fluid flow and heat transfer regimes were investigated to provide insight
into the phenomena that occur during subcooled flow boiling. The theory of each
regime was discussed in detail and followed by the development of a numerical model.
Numerical models to analyze natural convection, laminar flow, turbulent flow with and
without heat transfer, two-phase flow, pool boiling and subcooled flow boiling were
created. The commercial software Fluent® was used to produce the models and analyze
the results. Different modeling techniques and numerical solvers were employed in
Fluent depending on the scenario to generate acceptable results. The results of each
model were compared to experimental data when available to prove its validity.
Although numerous heat transfer and fluid flow phenomena were analyzed, the
primary focus of this research was subcooled flow boiling. The impact that different
boiling model options have on liquid volume fraction was examined. Three bubble
departure diameter models and two nucleation site density models were studied using the
same inlet conditions. The bubble departure diameter models examined did not show
any relationship with liquid volume fraction; however, the Kocamustafaogullari-Ishii
nucleation site density model tended to predict a greater liquid volume fraction, meaning
less vapor production, than the Lemmert-Chawla nucleation site density model.
A second study on how inlet conditions impact the liquid volume fraction during
subcooled flow boiling was explored.
The inlet conditions of heat flux, fluid
temperature and mass flow rate were increased or decreased relative to a base case value.
The difference in liquid volume fraction between scenarios was compared and
relationships relating the inlet conditions with respect to liquid volume fraction were
developed. Overall, the fluid temperature had the greatest impact on liquid volume
fraction, the wall heat flux had the second greatest impact and the mass flow rate had the
smallest impact.
xii
1. INTRODUCTION
Since the 19th century, the world’s standard of living has greatly increased
primarily due to the generation and distribution of electricity. Over 80% of the world’s
electricity production is generated by converting thermal energy, from a fuel source, into
electrical energy. The Rankine Cycle is a common energy conversion process that burns
fuel and generates steam which is used to spin an electric generator.
Electricity
production involves several engineering processes but is primarily based around heat
transfer and fluid flow.
Coal, oil, natural gas and uranium are some of the different fuel sources
available to electrical power plants. The fuel source in focus in this research is uranium
or nuclear fuel. Nuclear power plants harness energy released during fission to heat the
water that flows over the uranium fuel rods. The energy transfer mechanisms within a
nuclear reactor involve the three major forms of heat transfer; conduction, convection
and radiation. The fluid flow through the reactor is complex because of intense energy
transfer and phase change. In Pressurizer Water Reactors, the water flowing through the
reactor is prevented from bulk boiling because it is highly pressurized; however, a small
amount of localized boiling does occur which is known as subcooled flow boiling. This
research focuses on the convective heat transfer and fluid flow phenomena that occur
during subcooled flow boiling. Specifically, topics on turbulence, two-phase flow and
phase change are discussed.
Subcooled boiling occurs when an under-saturated fluid comes in contact with a
surface that is hotter than its saturation temperature. Small bubbles form on the heated
surface at preferential locations called nucleation sites. The number of bubbles that form
is heavily dependent on fluid temperature, pressure, mass flow, heat flux and
microscopic features of the surface. After the bubbles form on the heated surface, they
detach and enter the bulk fluid. When this occurs, saturated vapor is dispersed in a
subcooled liquid which is where the term subcooled boiling originates.
1
1.1
SUMMARY OF PRIOR WORK
Subcooled flow boiling is characterized by the combination of convection,
turbulence, boiling and two-phase flow. Determining the amount of voiding that occurs
during subcooled flow boiling has become a topic of great interest in recent years. A
number of mechanistic models for the prediction of wall heat flux and partitioning have
been developed. One of the most commonly used mechanistic models for subcooled
flow boiling was developed by Del Valle and Kenning. Their model accounts for bubble
dynamics at the heated wall using concepts developed initially by Graham and
Hendricks for wall heat flux partitioning during nucleate pool boiling. Recently, a new
approach to the partitioning of the wall heat flux has been proposed by Basu et al. The
fundamental idea of this model is that all of the energy from the wall is transferred to the
adjacent liquid.
A fraction of the energy is absorbed by vapor bubbles through
evaporation while the remainder goes into the bulk liquid. [1]
In addition to the development of mechanistic heat transfer and partitioning
models, focus has been placed on accurately modeling three of the most impactful
parameters in subcooled flow boiling. These parameters are the active nucleation site
density (Na), bubble departure diameter (dbw) and bubble departure frequency (f). The
two most common nucleation site density models were developed by Lemmert and
Chwala and Kocamustafaogullari and Ishii. Both of these models are available in
Fluent.
Many correlations have been developed to determine the bubble departure
diameter. Tolubinsky and Kostanchuk proposed the most simplistic correlation which
evaluates bubble departure diameter as a function of subcooling temperature.
Kocamustafaogullari and Ishii improved this model by including the contact angle of the
bubble. Finally, Unal produced a comprehensive correlation which includes the effect of
subcooling, the convection velocity and the heater wall properties. All three of these
bubble departure diameter correlations are available in Fluent.
The most common bubble departure frequency correlation for computational
fluid dynamics was developed by Cole. It is based on a bubble departure diameter
model and a balance between buoyancy and drag forces. The Cole bubble departure
frequency model is available in Fluent.
2
Recently, the use of population balance equations have been used to improve the
modeling of subcooled flow boiling by determining how swarms of bubbles interact
after detaching from the heated surface. This technique was recommended by Krepper
et. al. [2] and investigated by Yeoh and Tu [1]. Population balance equations have been
introduced in several branches of modern science, mainly areas with particulate entities
such as chemistry and materials because they help define how particle populations
develop in specific properties over time. Population balance equations are available in
Fluent; however, not in combination with the boiling model.
1.2
CONTENT
This research produced an investigation on subcooled flow boiling using Fluent.
Fluent is a widely accepted commercial computational fluid dynamics code that can
simulate complex heat transfer and fluid flow regimes. This thesis had three major
objectives. The first objective was to gain an understanding of the phenomena that occur
during subcooled flow boiling. The second objective was to determine how the boiling
model options described in Section 1.1 impact the liquid volume fraction at different
axial locations. The third objective was to evaluate how heat flux, fluid temperature and
mass flow rate impact the liquid volume fraction at different axial locations.
Due to its complexity, development of the subcooled flow boiling model was
performed in stages. With the expansion of each model, a more complicated fluid flow
or heat transfer scenario was analyzed. After each model was created, a mesh validation
was performed and the results were compared to known experimental data when
possible to validate the information generated by Fluent.
The first and simplest model created was for natural convection. The theory of
natural convection is described in Section 2.3 and the analytical modeling results are
presented in Section 3.1. Two natural convection geometries were analyzed. The first
was a horizontal cylinder suspended in an infinite pool and the second was a vertical
plate suspended in an infinite pool. The second model developed was for laminar flow.
The theory of laminar flow is described in Section 2.4 and the analytical modeling
results are discussed in Section 3.2. The third model developed was for turbulent flow.
3
The theory of turbulent flow is described in Section 2.5 and the analytical modeling
results are displayed in Section 3.3. Section 3.3 contains two turbulent flow scenarios;
turbulent flow without heat transfer and turbulent flow with heat transfer. The fourth
model developed was for two-phase flow with water and air. The theory of two-phase
flow is described in Section 2.6 and the analytical modeling results for the scenarios
analyzed are shown in Section 3.4. The first scenario is a gas mixing tank and the
second scenario is a bubble column. The final and most complex models created include
phase transformation (vaporization and condensation). Section 2.7 contains the theory
of boiling heat transfer with a subsection specific to subcooled boiling.
Section 3.5
presents the analytical results for the two models created; the first for pool boiling and
the second for subcooled flow boiling. A summary of the results and the conclusions
reached from the models developed herein is documented in Section 4.
4
2. HEAT TRANSFER AND FLUID FLOW: THEORY
This section discusses basic theory behind some common heat transfer and fluid
flow scenarios. It is meant to provide a brief introduction to the phenomena involved in
subcooled flow boiling.
2.1
GOVERNING EQUATIONS
Conservation equations are a local form of conservation laws which state that
mass, energy and momentum as well as other natural quantities must be conserved. A
number of physical phenomena may be described using these equations [3]. In fluid
dynamics, the two key conservation equations are the conservation of mass and the
conservation of momentum.
Conservation of Mass (continuity equation):
𝜕𝜌
⃑⃑ ∙ 𝜌v
+ (∇
⃑⃑) = 0
𝜕𝑡
Conservation of Momentum:
𝜌
𝐷v
⃑⃑
⃑⃑𝜌 + 𝜇∇
⃑⃑2 v
= −∇
⃑⃑ + 𝜌𝑔
𝐷𝑡
In subcooled flow boiling, as in many other instances of fluid dynamics, energy
is added or removed from the system. When this occurs, the conservation of energy
equation is important.
Conservation of Energy:
𝜌𝐶̂𝑝
𝐷𝑇
𝜕 ln 𝜌 𝐷𝑝
⃑⃑ ∙ 𝑞⃑) − (
= −(∇
)
𝐷𝑡
𝜕 ln 𝑇 𝑝 𝐷𝑡
5
2.2
NUMERICAL METHODS
After the conservation laws governing heat transfer, fluid flow and other related
processes are expressed in differential form (Section 2.1), they can solved using
numerical methods to determine pressure, temperature, mass flux, etc. for various
circumstances and boundary conditions.
Each differential equation represents a
conservation principle and employs a physical quantity as its dependent variable that is
balanced by the factors that influence it. Some examples of differential equations that
may be solved through numerical methods are conservation of energy, conservation of
momentum and time-averaged turbulent flow. [4]
The goal of computational fluid dynamics is to calculate the temperature,
velocity, pressure, etc. of a fluid at particular locations within a system. Thus, the
independent variable in the differential equations is a physical location (and time in the
case of unsteady flows). Due to computational limitations, the number of locations (also
known as grid points or nodes) must be finite. By concentrating on a solution to the
differential equations at discrete locations, the requirement to find an exact solution is
avoided. The algebraic equations (also known as discretization equations) involving the
unknown values of the independent variable at chosen locations (grid points) are derived
from the differential equations governing the independent variable. In this derivation,
assumptions about the value of the independent variable between grid points must be
made. This concept is known as discretization. [4]
A discretization equation is an algebraic relationship that connects the values of
the dependent variable for a group of grid points within a control volume. This type of
equation is derived from the differential equation governing the dependent variable and
thus expresses the same physical information as the differential equation. The piecewise
nature of the profile (or mesh) is created by the finite number of grid points that
participate in a given discretization equation. The value of the dependent variable at a
grid point thereby influences the value of the dependent variable in its immediate area.
As the number of grid points becomes very large, the solution of the discretization
equations is expected to approach the exact solution of the corresponding differential
equation. This is true because as the grid points get closer together, the change in value
between neighboring grid points becomes small and the actual details of the profile
6
assumption become less important.
This is where the term “mesh independent”
originates. If there are too few grid points (coarse mesh), the profile assumptions can
impact the solution results and the discretization equation solution will not match the
differential equation solution. To ensure that the discretization equation results are not
dependent on the profile assumptions, the solution should be checked for mesh
independence. [4]
One of the more common procedures for deriving discretization equations is
using a truncated Taylor series. Other methods include variational formulation, method
of weighted residuals and control volume formulation. The conservation equations in
Section 2.1 in discretized form are shown below:
Pressure Correction Equation (continuity equation) [4]:
′
𝑎𝑃 𝑝𝑃′ = 𝑎𝐸 𝑝𝐸′ + 𝑎𝑊 𝑝𝑊
+ 𝑎𝑁 𝑝𝑁′ + 𝑎𝑆 𝑝𝑆′ + 𝑎 𝑇 𝑝𝑇′ + 𝑎𝐵 𝑝𝐵′ + 𝑏
𝑎𝐸 = 𝜌𝑒 𝑑𝑒 ∆𝑦 ∆𝑧
𝑎𝑊 = 𝜌𝑤 𝑑𝑤 ∆𝑦 ∆𝑧
𝑎𝑁 = 𝜌𝑛 𝑑𝑛 ∆𝑧 ∆𝑥
𝑎𝑆 = 𝜌𝑠 𝑑𝑠 ∆𝑧 ∆𝑥
𝑎 𝑇 = 𝜌𝑡 𝑑𝑡 ∆𝑥 ∆𝑦
𝑎𝐸 = 𝜌𝑒 𝑑𝑒 ∆𝑥 ∆𝑦
𝑏=
(𝜌𝑃0 − 𝜌𝑃 )∆𝑥 ∆𝑦 ∆𝑧
+ [(𝜌𝑢∗ )𝑤 − (𝜌𝑢∗ )𝑒 ]∆𝑦 ∆𝑧 + [(𝜌𝑣 ∗ )𝑠 − (𝜌𝑣 ∗ )𝑛 ]∆𝑧 ∆𝑥
∆𝑡
+ [(𝜌𝑤 ∗ )𝑏 − (𝜌𝑤 ∗ )𝑡 ]∆𝑥 ∆𝑦
Figure 2.2-1: Control Volume Schematic for Pressure Correction Equation
7
Conservation of Momentum in Discretized Form [4]:
∗
𝑎𝑒 𝑢𝑒∗ = ∑ 𝑎𝑛𝑏 𝑢𝑛𝑏
+ 𝑏 + (𝑝𝑃∗ − 𝑝𝐸∗ )𝐴𝑒
∗
𝑎𝑛 𝑣𝑛∗ = ∑ 𝑎𝑛𝑏 𝑣𝑛𝑏
+ 𝑏 + (𝑝𝑃∗ − 𝑝𝑁∗ )𝐴𝑛
(a)
(b)
Figure 2.2-2: Control Volume Schematic for Momentum Equation
Conservation of Energy in Discretized Form [4]:
𝑎𝑃 ɸ𝑃 = 𝑎𝐸 ɸ𝐸 + 𝑎𝑊 ɸ𝑊 + 𝑎𝑁 ɸ𝑁 + 𝑎𝑆 ɸ𝑆 + 𝑏
𝑎𝐸 = 𝐷𝑒 𝐴(|𝑃𝑒 |) + ⟦−𝐹𝑒 , 0⟧
𝑎𝑊 = 𝐷𝑤 𝐴(|𝑃𝑤 |) + ⟦𝐹𝑤 , 0⟧
𝑎𝑁 = 𝐷𝑛 𝐴(|𝑃𝑛 |) + ⟦−𝐹𝑛 , 0⟧
𝑎𝑆 = 𝐷𝑠 𝐴(|𝑃𝑠 |) + ⟦𝐹𝑠 , 0⟧
𝑎𝑃0 =
𝜌𝑃0 ∆𝑥 ∆𝑦
∆𝑡
𝑏 = 𝑆𝐶 ∆𝑥 ∆𝑦 + 𝑎𝑃0 ɸ0𝑃
𝑎𝑃 = 𝑎𝐸 + 𝑎𝑊 + 𝑎𝑁 + 𝑎𝑆 + 𝑎𝑃0 − 𝑆𝑃 ∆𝑥 ∆𝑦
Figure 2.2-3: Control Volume Schematic for Energy Equation
8
In the iterative process for solving a discretization equation, it is often desirable
to speed up or to slow down the changes, from iteration to iteration, in the values of the
dependent variable. The process of accelerating the rate of change between iterations is
called over-relaxation while the process of slowing down the rate of change between
iterations is called under-relaxation. To avoid divergence in the iterative solution of
strongly nonlinear equations, under-relaxation is a very useful tool [4].
Fluent allows for manipulation of the relaxation constants for many independent
variables to improve convergence ability. It also offers numerous spatial discretization
solvers for the various independent variables such as pressure, flow, momentum,
turbulence, and energy.
Fluent implements the control volume formulation with
upwinding which was first proposed by Courant, Isaacson, and Rees in 1952. Other
options include QUICK, power law and third-order MUSCL.
2.3
NATURAL CONVECTION
Convection is the transport of mass and energy by bulk fluid motion. If the fluid
motion is induced by some external force, like a pump, fan, or suction device, it is
generally referred to as forced convection. If the fluid motion is induced by an internal
force such as buoyancy produced by density gradients, it is generally referred to as
natural convection. The density gradients can arise from mass concentration and or
temperature gradients in the fluid [5]. For example, in a system where a heated surface
is in contact with a cooler fluid, the cooler fluid absorbs energy from the heated surface
and becomes less dense. Buoyancy effects due to body forces cause the heated fluid to
rise and the surrounding, cooler fluid takes its place. The cooler fluid is then heated and
the process continues forming a convection cell that continuously removes energy from
the heated surface.
In nature, natural convection cells occur everywhere from oceanic currents to air
rising above sunlight-warmed land.
Natural convection also takes place in many
engineering applications such as home heating radiators and cooling of computer chips.
The amount of heat transfer that occurs due to natural convection in a system is
characterized by the Grashof, Prandtl and Rayleigh numbers.
9
The Grashof number,
Gr, is a dimensionless parameter that represents the ratio of buoyancy to viscous forces
acting on a fluid and is defined as:
𝐺𝑟 =
𝑔𝛽(𝑇𝑤𝑎𝑙𝑙 − 𝑇∞ )𝐿3
(𝜇 ⁄𝜌)2
where β is the thermal expansion coefficient:
1 𝜕𝜌
𝛽=− ( )
𝜌 𝜕𝑇 𝑝
The Prandtl number, Pr, is a dimensionless parameter that represents the ratio of
momentum diffusivity to thermal diffusivity; and is defined as:
Pr =
Cp μ
k
The Rayleigh number, Ra, is a dimensionless parameter that represents the ratio
of buoyancy to viscosity forces times the ratio of momentum diffusivity to thermal
diffusivity; and is defined as:
Ra = GrPr
When the Rayleigh number is below a critical value for a particular fluid, heat
transfer is primarily in the form of conduction; when it exceeds the critical value, heat
transfer is primarily in the form of convection.
Like forced convection, natural
convection can either be laminar or turbulent. Rayleigh numbers less than 108 indicate a
buoyancy-induced laminar flow, with transition to turbulence occurring at about 109. [6]
In many situations, convection is mixed meaning that both natural and forced
convection occur simultaneously.
The importance of buoyancy forces in a mixed
convection flow can be measured by the ratio of the Grashof and Reynolds numbers:
Gr
gβ(𝑇𝑤𝑎𝑙𝑙 − 𝑇∞ )L
=
Re2
V2
When this ratio approaches or exceeds unity, there are strong buoyancy
contributions to the flow. Conversely, if the ratio is very small, buoyancy forces may be
ignored.
10
2.4
LAMINAR FLOW
Fluid flow can be grouped into two categories, laminar or turbulent flow.
Laminar flow implies that the fluid moves in sheets that slip relative to each other and it
occurs at very low velocities where there are only small disturbances and little to no
local velocity variations. In laminar flow, the motion of the fluid particles is very
orderly and can be characterized by high momentum diffusion and low momentum
convection.
The Reynolds number is used to characterize the flow regime. The Reynolds
number, Re, is a dimensionless number that represents the ratio of inertial forces to
viscous forces; and is defined as:
Re =
ρVL
μ
The Reynolds number helps quantify the relative importance of inertial and
viscous forces for given flow conditions. For internal flow, such as within a pipe,
laminar flow occurs at a Reynolds number less than 2300.
The velocity profile of a laminar flow in a pipe can be calculated by [5]:
𝑢𝑥 =
𝑟𝑠2
𝑑𝑝
𝑟2
(− ) (1 − 2 )
4𝜇
𝑑𝑥
𝑟𝑠
Or, in terms of the mean velocity, V:
𝑟2
𝑢𝑥 = 2𝑉 (1 − 2 )
𝑟𝑠
The above two equations indicate that the velocity for laminar flow is related to the
square of the pipe radius and thus the flow profile is parabolic.
The energy equation for flow through a circular pipe assuming symmetric heat
transfer, fully developed flow and constant fluid properties is [5]:
𝜕𝑇
1𝜕
𝜕𝑇
𝜕 2𝑇
𝑢𝑥
= 𝛼[
(𝑟 ) + 2 ]
𝜕𝑥
𝑟 𝜕𝑟 𝜕𝑟
𝜕𝑥
This equation shows that convection due to flow is balanced by diffusion in the radial
and axial directions.
11
2.5
TURBULENT FLOW
In fluid dynamics, turbulence is a flow regime characterized by chaotic and
stochastic changes. Turbulent flows involve large Reynolds numbers and contain threedimensional vorticity fluctuations. The unsteady vortices appear on many scales and
interact with each other generating high levels of mixing and increased rates of
momentum, heat and mass transfer. Like laminar flows, turbulent flows are dissipative
and therefore depend on their environment to obtain energy. A common source of
energy for turbulent velocity fluctuations is shear in the mean flow; other sources, such
as buoyancy, exist too. If turbulence arrives in an environment where there is no shear
or other maintenance mechanism, the turbulence decays and the flow tends to become
laminar. [7]
In flows that are originally laminar, turbulence arises from instabilities at large
Reynolds numbers.
For internal flows, such as within a pipe, turbulent flow is
characterized by a Reynolds number greater than 4000. For flows with a Reynolds
number between 2300 and 4000, both laminar and turbulent flows are possible. This is
called transition flow. [7]
A common example of the transition from laminar flow to turbulent flow is
smoke rising from a cigarette [8].
Figure 2.5-1: Transition from Laminar to Turbulent Flow
12
As the smoke leaves the cigarette, it travels upward in a laminar fashion as
shown by the single stream of smoke. At a certain distance, the Reynolds number
becomes too large and the flow begins to transition to the turbulent regime. When this
happens, the flow of the smoke becomes more random and rapidly mixes with the air
causing it to dissipate.
Modeling of turbulent flow requires the exact solution of the Continuity and
Navier-Stokes equations which can be extremely difficult and time consuming due to the
many scales involved. To reduce the complexity, an approximation to the Navier-Stokes
equations was developed by Osborne Reynolds called the Reynolds-averaged Navier–
Stokes equations (or RANS equations). This method decomposes the instantaneous
fluid flow quantities of the Navier-Stokes equations into mean (time-averaged) and
fluctuating components. The RANS equations can be used with approximations based
on knowledge of the turbulent flow to give approximate time-averaged solutions to
the Navier–Stokes equations. [9]
For the velocity terms:
𝑢 = 𝑢̅ + 𝑢′
where 𝑢̅𝑖 and 𝑢𝑖′ are the mean and fluctuating velocity components respectively.
Similarly, for scalar quantities:
𝜙 = 𝜙̅ + 𝜙 ′
where 𝜙 denotes a scalar such as energy, pressure, or species concentration.
Substituting expressions of this form for the flow variables into the instantaneous
continuity and momentum equations and taking a time-average yields the time-averaged
continuity and momentum equations [9]. These are written in Cartesian tensor form as:
𝜕𝑝
𝜕
(𝜌𝑢̅𝑖 ) = 0
+
𝜕𝑡 𝜕𝑥𝑖
𝜕
𝜕
𝜕𝑝
𝜕
𝜕𝑢𝑖 𝜕𝑢𝑗 2 𝜕𝑢𝑙
𝜕
′ ′
̅̅̅̅̅̅
(𝜌𝑢̅𝑖 ) +
(𝜌𝑢̅𝑖 𝑢̅𝑗 ) = −
+
[𝜇 (
+
− 𝛿𝑖𝑗
)] +
(−𝜌𝑢
𝑖 𝑢𝑗 )
𝜕𝑡
𝜕𝑥𝑗
𝜕𝑥𝑖 𝜕𝑥𝑗
𝜕𝑥𝑗 𝜕𝑥𝑖 3
𝜕𝑥𝑙
𝜕𝑥𝑗
The two above equations are the Cartesian RANS equations for a twodimensional system. They have the same general form as the instantaneous NavierStokes equations, with the velocities and other solution variables now representing time13
averaged values. The RANS equations can be used with approximations based on
knowledge of the turbulent flow to give approximate time-averaged solutions to
′ ′
̅̅̅̅̅̅
the Navier–Stokes equations. An additional term (−𝜌𝑢
𝑖 𝑢𝑗 ), known as the Reynolds
stress, appears in the equation as a results of using the RANS method. [9]
One way that the Reynolds stress is evaluated in practice is through the k-ϵ
turbulence model. The k-ϵ model was first introduced by Harlow and Nakayama in
1968 [10]. The k-ϵ model has become the most widely used model for industrial
applications because of its overall accuracy and small computational demand. In the k-ϵ
model, k represents the turbulent kinetic energy and ϵ represents its dissipation rate.
Turbulent kinetic energy is the average kinetic energy per unit mass associated with
eddies in the turbulent flow while epsilon (ϵ) is the rate of dissipation of the turbulent
energy per unit mass.
In the derivation of the k-ϵ model, it is assumed that the flow is fully turbulent,
and the effects of molecular viscosity are negligible. As the strengths and weaknesses of
the standard k-ϵ model have become known, modifications were introduced to improve
its performance. These improvements have helped create many, new, more accurate
models, among them, the realizable k-ϵ model which differs from the standard k-ϵ model
in two important ways. First, the realizable model contains an alternative formulation of
the turbulent viscosity. Second, a modified transport equation for the dissipation rate, ϵ,
is derived from an exact equation for the transport of the mean-square vorticity
fluctuation. The term “realizable” means that the model satisfies certain mathematical
constraints on the Reynolds stresses, consistent with the physics of turbulent flow. [9]
2.5.1
CALCULATING TURBULENCE PARAMETERS
All of the computational fluid dynamic models discussed in this thesis use the
k-ϵ turbulence model when applicable. In Fluent, turbulence models require certain
parameters to be established prior to initialization to properly set the boundary
conditions for the flow.
Based on the conditions specified in Table 2.5.1-1, the
equations in Table 2.5.1-2 [9] were used to determine the boundary condition inputs for
the turbulent flow models presented in Section 3.3.
14
Table 2.5.1-1: Turbulent Flow Input
Input Parameter
Mass Flow Rate (𝑚̇)
Pipe Diameter (D)
Viscosity (μ)
Density (ρ)
Turbulence Empirical Constant (Cμ)
Numerical Value
1.0 kg/s
0.03 m
0.001003 kg/m-s
998.2 kg/m3
0.09 [9]
Table 2.5.1-2: Calculation of Turbulent Parameters
Variable
Equation
Numerical Value
4∗𝐴
𝑃
𝐷 2
𝜋 ∗ (2 )
=
=𝐷
4∗𝜋∗𝐷
𝐷 2
𝐴 =𝜋∗( )
2
0.03 𝑚 2
=𝜋∗(
)
2
𝑚̇
𝑉=
𝜌∗𝐴
0.5 𝑘𝑔/𝑠
=
𝑘𝑔
998.2 3 ∗ 0.00070686 𝑚2
𝑚
𝑚̇𝐷ℎ
𝑅𝑒𝐷ℎ =
𝜇𝐴
𝑘𝑔
0.5 𝑠 ∗ 0.03 m
=
𝑘𝑔
0.001003 𝑚 − 𝑠 ∗ 0.00070686 𝑚2
𝑙 = 0.07 ∗ 𝐷ℎ
= 0.07 ∗ 0.03 𝑚
𝐷ℎ =
Hydraulic Diameter (Dh)
Flow Area (A)
Average Flow Velocity (V)
Reynolds Number (ReDh)
Turbulent Length Scale (l)
Turbulent Intensity (I)
𝐼=
1
−
0.16 ∗ 𝑅𝑒𝐷ℎ8
−
Turbulent Kinetic Energy (k)
= 0.16 ∗ 42314
3
2
𝑘 = (𝑢𝑎𝑣𝑔 ∗ 𝐼)
2
2
3
𝑚
= (1.41726 ∗ 0.0422483)
2
𝑠
3/4 k
Dissipation Rate (ϵ)
1
8
ε = C𝜇
0.00070686 m2
1.41726 m/s
42314
0.0021 m
0.0422483
0.0053785 m2/s2
3/2
𝑙
3/2
0.0053785
= 0.093/4
0.0021
15
0.03 m
0.030859 m2/s3
2.6
TWO-PHASE FLOW
Fluid flow that contains two or more components is referred to as multiphase
flow. The flow components can be of the same chemical substance but in different
states of matter such as water and steam, be of different chemical substances but the
same state of matter such as water and oil or finally be of different chemical substance
and different states of matter such as water and air. This section focuses on two-phase
flow involving water and air while Section 2.7 focuses on two-phase flow involving
water and steam.
Depending on the volume fraction of each component in the two-phase flow,
different flow patterns can exist. Understanding the two-phase flow pattern is important
because pressure drops and heat transfer rates are heavily impacted by the flow type.
The characteristic flow patterns for two-phase flow, in order of increasing gas volume
fraction from liquid to gas, are bubbly flow, plug flow, stratification flow, wavy flow,
slug flow, annular flow and spray flow. A schematic representation of each of these
flow patterns is shown in Figure 2.6-1 [11].
Figure 2.6-1: Two-Phase Flow Patterns
The flow patterns shown in Figure 2.6-1 can be further classified into three
categories, bubbly flow, slug flow and annular flow. Bubbly flow is when the liquid
phase is continuous and the vapor phase is discontinuous such that the vapor phase is
16
distributed in the liquid phase in the form of bubbles. This flow pattern occurs at low
gas volume fractions. Subcooled flow boiling is classified as bubbly flow. Slug flow is
when there are relatively large liquid slugs surrounded by vapor. This flow pattern
occurs at moderate gas volume fractions and relatively low flow velocities. Annular
flow is when the liquid phase is continuous along the wall and the vapor phase is
continuous in the core. This flow pattern occurs at high gas volume fractions and high
flow velocities. Although not considered to be a flow regime, flow film boiling is the
opposite of annular flow (the vapor phase is continuous along the wall and the liquid
phase is continuous in the core) and occurs when the heat flux is relatively large
compared to the mass flux. Film boiling is discussed further in Section 2.7.
The flow pattern of a system can be determined using the Baker flow criteria
shown in Figure 2.6-2 [11] if the gas volume fraction and mass velocity are known. For
example, if a two-phase flow consisting of air and water has a total mass velocity (air
plus water) of 0.10 x 106 lbm/hr-ft2 and a gas quality of 0.4, then flow will be annular.
Figure 2.6-2: Baker Flow Pattern
17
2.6.1
MODELING TWO-PHASE FLOW
Two-phase flows obey the same basic laws of fluid mechanics that apply to
single phase flows; however, the equations are more complicated and more numerous.
Two-phase flows are more difficult to solve due to the secondary phase and additional
phenomena that must be accounted for such as mass transfer and phase-interface
interactions (slip and drag). Three common multiphase flow models available in Fluent
are Volume of Fluid (VOF), Mixture and Eulerian, each with varying strengths and
computational demands.
The VOF model is the simplest and least computationally expensive of the three
multiphase models offered in Fluent.
The VOF model can analyze two or more
immiscible fluids by solving a single set of momentum equations and tracking the
volume fraction of each fluid throughout the domain. All control volumes must be filled
with either a single fluid phase or a combination of phases. The VOF model does not
allow for void regions where no fluid of any type is present. The VOF method was
based on the marker-and-cell method and quickly became popular due to its low
computer storage requirements. Typical applications of VOF include stratified or freesurface flows such as the prediction of jet breakup, the motion of large bubbles in a
liquid, the motion of liquid after a dam break, and the steady or transient tracking of a
liquid-gas interface. [9]
The Mixture model is between the VOF and Eulerian multiphase models both in
complexity and computational expense. The Mixture model can analyze multiple phases
(fluid or particulate) by solving the momentum, continuity, and energy equations for the
mixture, the volume fraction equations for the secondary phases, and algebraic
expressions for the relative velocities. Like the VOF model, it uses a single-fluid
approach but has two major differences. First, the Mixture model allows for the phases
to be interpenetrating and therefore the volume fraction of a fluid in a control volume
can be equal to any value between zero and one. Second, the Mixture model allows for
the phases to move at different velocities, using the concept of slip. The Mixture model
is a good substitute for the full Eulerian model in several cases where a full multiphase
model may not be feasible or when the interphase laws are unknown or their reliability
can be questioned.
Typical applications include sedimentation, cyclone separators,
18
particle-laden flows with low loading, and bubbly flows where the gas volume fraction
remains low. [9]
The Eulerian model is the most complex and most computationally expensive
multiphase model offered in Fluent.
It solves the momentum and the continuity
equations for each phase, and couples the equations through pressure and exchange
coefficients. With the Eulerian model, the number of secondary phases is limited only
by memory requirements and convergence behavior. The Eulerian model allows for the
modeling of multiple separate, yet interacting phases. The interacting phases can be
liquids, gases, or solids in nearly any combination.
Due to its ability to model
interacting phases, typical applications of the Eulerian model are bubble columns, risers,
particle suspension, fluidized beds and boiling including subcooled boiling. [9]
2.6.2
POPULATION BALANCE MODEL
In many two-phase flow applications, including subcooled flow boiling, it is
helpful to know how the secondary phase (solids, bubbles, droplets, etc.) evolves over
time. Thus, a balance equation is required to describe the changes in the particle size
distribution over time, in addition to the momentum, mass, and energy balances already
employed. The additional balance equation is generally referred to as the population
balance equation.
The population balance model in Fluent implements a number density function to
account for the different sizes of the particle population.
With the aid of particle
properties (i.e., size, density, porosity, composition, etc.), different particles in the
population can be distinguished and their behavior can be described. [9]
The link between the population balance and boiling models has not been fully
developed in Fluent and is therefore not employed in the subcooled flow boiling model
discussed in Section 3.5.2. However, the population balance model is utilized to track
bubble size distribution within a bubble column (Section 3.4.3).
19
2.7
BOILING HEAT TRANSFER
Boiling is a mode of heat transfer that occurs when saturated liquid changes to
saturated vapor due to heat addition. It is normally characterized by a high heat transfer
capacity and a low wall-fluid temperature delta which is made possible by the generally
large energy absorption required to cause a phase change. These heat transfer properties
are essential in industrial cooling applications such as nuclear reactors and fossil boilers.
Because of its importance in industry, a significant amount of research has been carried
out to study the capacity and the mechanism of boiling heat transfer. There are two
basic types of boiling, pool boiling and flow boiling. If heat addition causes a phase
change in a stagnant fluid, then it is called pool boiling. If heat addition causes a phase
change in a moving fluid, then it is called flow boiling. Both types of boiling heat
transfer can be separated into four regimes, which are shown in Figure 2.7-1 [12].
Figure 2.7-1: Boiling Heat Transfer Regimes
The first regime of boiling, up to point A, is known as natural convection boiling.
During this regime, no bubbles form; instead, heat is transferred from the surface to the
5/4
bulk fluid by natural convection. The heat transfer rate is proportional to 𝛥𝑇𝑠𝑎𝑡 [11].
The second regime of boiling, from point A to point C, is called nucleate boiling.
During this stage, vapor bubbles are generated at certain preferred locations on the
heated surface called nucleation sites. Nucleation sites are often microscopic cavities or
cracks in the surface. When the liquid near the wall superheats, it evaporates and a
20
significant amount of energy is removed from the heated surface due to the latent heat of
the vaporization which also increases the convective heat transfer by mixing the liquid
water near the heated surface. There are two subregimes of nucleate boiling that can
take place between points A and C. The first subregime is when local boiling occurs in a
subcooled liquid (subcooled boiling). In this situation, bubbles form on a heated surface
but tend to condense after detaching from it. The second subregime is when local
boiling occurs in a saturated liquid. In this case, bubbles do not condense after detaching
from the heated surface since the liquid is at the same temperature as the vapor.
Nucleate boiling is characterized by a very high heat transfer rate and a small
temperature difference between the bulk fluid and the heated surface. For this reason, it
is considered to be the most efficient form of boiling heat transfer. [11]
As the heated surface increases in temperature, more and more nucleation sites
become active. As more bubbles form at these sites, they begin to merge together and
form columns or slugs of vapor, thus decreasing the contact area between the bulk fluid
and the heated surface. The decrease in contact area causes the slope of the line in
Figure 2.7-1 to decrease until a maximum is reached (point C). Point C is referred to as
the critical heat flux and the vapor begins to form an insulating blanket around the
heated surface which dramatically increases the surface temperature when reached. This
is called the boiling crisis or departure from nucleate boiling. [12]
As the temperature delta increases past the critical heat flux, the rate of bubble
generation exceeds the rate of bubble separation. Bubbles at the different nucleation
sites begin to merge together and boiling becomes unstable. The surface is alternately
covered with a vapor blanket and a liquid layer, resulting in oscillating surface
temperatures. This regime of boiling is known as partial film boiling or transition
boiling and takes place between points C and D. [11]
If the temperature difference between the surface and the fluid continues to
increase, stable film boiling is achieved.
During stable film boiling, there is a
continuous vapor blanket surrounding the heated surface and phase change occurs at the
liquid-vapor interface instead of at the heated surface. During this regime, most heat
transfer is carried out by radiation. [12]
21
2.7.1
SUBCOOLED BOILING
Subcooled flow boiling occurs when a moving, under-saturated fluid comes in
contact with a surface that is hotter than its saturation temperature. Intense interaction
between the liquid and vapor phases occur and therefore the Eulerian multiphase model
is most appropriate for subcooled boiling because it is capable of modeling multiple,
separate, yet interacting phases.
The heat transfer rate from the wall to the fluid changes based on the amount of
vapor on the heated surface. Since the vapor area is constantly changing due to the
formation, growth and departure of bubbles, the use of a correlation is necessary. Del
Valle and Kenning created a mechanistic model to determine the area of the heated
surface influenced by vapor during flow boiling which is utilized by Fluent. When
modeling subcooled boiling, there are three parameters of importance that greatly impact
the liquid volume fraction; they are active nucleation site density (Na), bubble departure
diameter (dbw) and bubble departure frequency (f) [1].
As discussed previously, nucleation sites are preferential locations where vapor
tends to form and are usually cavities or irregularities in a heated surface. The number
of active nucleation sites per unit area is dependent on fluid and surface conditions. The
most common active nucleation site density relationship was developed by Lemmert and
Chwala. It is based on the heat flux partitioning data generated by Del Valle and
Kenning [1]:
𝑁𝑎 = [𝑚(𝑇𝑠𝑎𝑡 − 𝑇𝑤𝑎𝑙𝑙 )]𝑛
According to Kurul and Podowski, the values of m and n are 210 and 1.805,
respectively. Another popular correlation for nucleation site density was created by
Kocamustafaogullari and Ishii. They assumed that the active nucleation site density
correlation developed for pool boiling could be used in forced convective systems if the
effective superheat was used rather than the actual wall superheat. This correlation
accounts for both the heated surface conditions and the fluid properties [1]:
𝑁𝑎 =
1
2
𝑑𝑏𝑤
[∆𝑇
2𝜎𝑇𝑠𝑎𝑡
𝑒𝑓𝑓 𝜌𝑔 ℎ𝑓𝑔
−4.4
]
𝑓(𝜌∗ )
𝑓(𝜌∗ ) = 2.157 ∗ 10−7 ∗ 𝜌∗−3.2 ∗ (1 + 0.0049𝜌∗ )
22
𝜌∗ = (
𝜌𝑙 −𝜌𝑔
𝜌𝑔
)
∆𝑇𝑒𝑓𝑓 = 𝑆(𝑇𝑠𝑎𝑡 − 𝑇𝑤𝑎𝑙𝑙 )
The bubble departure diameter is the bubble size when it leaves the heated
surface and it depends in a complex manner on the amount of subcooling, the flow rate,
and a balance of surface tension and buoyancy forces. Determining the lift off bubble
diameter is crucial because the bubble size influences the interphase heat and mass
transfer through the interfacial area and the momentum drag terms. Many correlations
have been proposed for this purpose; however, the following three are applicable for low
pressure, subcooled flow boiling. The first correlation was proposed by Tolubinsky and
Kostanchuk. It establishes the bubble departure diameter as a function of the subcooling
temperature [1]:
𝑑𝑏𝑤 = 𝑚𝑖𝑛 [0.0006 ∗ exp (−
𝑇𝑠𝑢𝑏
45
) ; 0.00014]
The second correlation was created by Kocamustafaogullari and Ishii who
modified an expression by Fritz that involved the contact angle of the bubble. Its basic
premise is to balance the buoyancy and surface tension forces at the heated surface [1]:
𝑑𝑏𝑤 = 2.5 ∗ 10−5 (
𝜌𝑙 − 𝜌𝑔
𝜎
) 𝜃√
𝜌𝑔
𝑔 ∗ (𝜌𝑙 − 𝜌𝑔 )
A third, more comprehensive correlation was proposed by Unal which includes
the effect of subcooling, the convection velocity, and the heated wall properties [1]:
𝑑𝑏𝑤 =
2.42 ∗ 10−5 ∗ 𝑝0.709 ∗ 𝑎
√𝑏𝛷
where
𝑎=
(𝑄𝑤 − ℎ𝑇𝑠𝑢𝑏 )1/3 𝑘𝑙
𝑘𝑤𝑎𝑙𝑙 𝜌𝑤𝑎𝑙𝑙 𝐶𝑝,𝑤𝑎𝑙𝑙
𝑇𝑠𝑢𝑏
;𝑏 =
√
1/3
𝑘𝑙 𝜌𝑙 𝐶𝑝,𝑙
2[1 − (𝜌𝑔 /𝜌𝑙 )]
2𝐶 ℎ𝑓𝑔 √𝜋𝑘𝑙 ⁄𝜌𝑙 𝐶𝑝,𝑙 𝜌𝑔
23
3
𝐶=
ℎ𝑓𝑔 𝜇𝑙 [𝐶𝑝,𝑙 ⁄(0.013ℎ𝑓𝑔 𝑃𝑟 1.7 )]
𝜎
√𝑔(𝜌 − 𝜌 )
𝑙
𝑔
(𝑢𝑙 ) 0.47
𝑓𝑜𝑟 𝑢𝑙 ≥ 0.61 𝑚/𝑠
Φ = {0.61
1.0
𝑓𝑜𝑟 𝑢𝑙 < 0.61 𝑚/𝑠
The bubble departure frequency is the rate at which bubbles are generated and
detach from an active nucleation site. The most common bubble departure frequency
correlation for computational fluid dynamics was developed by Cole who derived it
based on the bubble departure diameter and a balance between buoyancy and drag
forces [1]:
𝑓=√
4𝑔(𝜌𝑙 − 𝜌𝑔 )
3𝜌𝑙 𝑑𝑏𝑤
The use of a mechanistic heat transfer model with individual correlations to
calculate the number of active nucleation sites, the bubble departure diameter and the
bubble departure frequency assist in the accurate determination of liquid volume fraction
during subcooled flow boiling. Each of these correlations are tested and compared in
Section 3.5.2.
24
3. HEAT TRANSFER AND FLUID FLOW: MODELING
3.1
NATURAL CONVECTION
Two natural convection scenarios were examined.
The first was a heated
horizontal cylinder and the second was a heated vertical plate, both were submerged in
an infinite pool of liquid. These examples were chosen because of their simplicity,
because they are commonly found in nature and because they have been previously
studied and results are available for validation of the numerical computations.
3.1.1
HORIZONTAL CYLINDER
A cylinder with a constant surface temperature submerged in an infinite pool of
liquid at a lower temperature was analyzed. Energy passed from the slightly warmer
cylinder to the nearby fluid causing its temperature to increase and convection cells to
form. Figure 3.1.1-1 shows a schematic representation of the geometry and boundary
conditions used to model the horizontal cylinder. The top and bottom walls of the
rectangle represent inlet and outlet pressure boundaries respectively, with pressure
conditions set such that the fluid is stagnant until heated by the cylinder. The left and
right walls of the rectangle are slip boundaries to more accurately model an infinite pool.
See Table 3.1.1-1 for a detailed list of input parameters used.
Figure 3.1.1-1: Horizontal Cylinder Schematic
25
Table 3.1.1-1: Horizontal Cylinder Model Input
Input
Geometry
Cylinder Diameter
Pool Height
Pool Width
2D Space
Solver
Time
Time Step Size
Type
Velocity Formulation
Gravity
Models
Energy
Viscous
Density
Initial Conditions
Cylinder Surface Temperature
Fluid Temperature
Material Properties (Water)
Specific Heat
Thermal Conductivity
Viscosity
Density
Solution Methods
Scheme
Gradient
Pressure
Momentum
Energy
Transient Formulation
Value
0.02 m
0.28 m
0.24 m
Planar
Transient
0.05 s
Pressure Based
Relative
-9.8 m/s2 (Y-direction)
Active
Laminar
Boussinesq
310 K
300 K
4182 J/kg-K
0.6 W/m-K
0.001003 kg/m-s
See Table 3.1.1-2
PISO
Least Square Cell Based
PRESTO!
Second Order Upwind
Second Order Upwind
Second Order Implicit
Table 3.1.1-2: Horizontal Cylinder Model Fluid Density
Density (kg/m3)
999.9
994.1
974.9
958.4
Temperature (K)
273
308
348
373
26
Figure 3.1.1-2 presents the liquid temperature field after 20 seconds of heating.
As the temperature increases, the fluid begins to rise due to buoyancy forces.
Figure 3.1.1-2: Temperature (K)
Figure 3.1.1-3 shows that even the fluid not in direct contact with the heated
cylinder experiences a density change. The density gradient which is caused by energy
transfer via conduction to the bulk fluid is illustrated by the color transition surrounding
the cylinder from least dense (blue) to most dense (red).
Figure 3.1.1-3: Density (kg/m3)
27
As the warm fluid rises, it loses energy to the surrounding bulk fluid which
reduces its buoyancy driving head until the rising fluid eventually stops. When the fluid
reaches its maximum elevation, it is pushed aside by the fluid travelling upwards below
it and begins to sink. This motion creates a small convection cell to the left and to the
right of the rising plume about 3 cm above the heated cylinder. This process continues
as long as there is a temperature gradient between the cylinder and the bulk fluid. If the
bulk fluid temperature increases, the buoyancy driving head will be smaller and the
convection cells will develop closer to the heated surface.
Figure 3.1.1-4 is a velocity vector plot that displays how the liquid moves within
the control volume. The two convection cells above the cylinder are clearly visible in
this figure which also reveals how the rising fluid is replaced by the cooler fluid
surrounding the cylinder.
Figure 3.1.1-4: Velocity Vectors (m/s)
To verify that the model produced realistic results, the solution was compared to
experimental data.
Figure 3.1.1-5 shows interference fringes surrounding a heated
horizontal cylinder in natural convection. Each interference fringe can be interpreted as
a band of constant density and therefore temperature.
28
(a)
(b)
Figure 3.1.1-5: Interference Fringes Around a Heated Horizontal Cylinder
(a) From Eckert [13] (b) Isotherms From Fluent
Figure 3.1.1-5 shows that the experimental data and the model solution have
isotherms that extend away from the cylinder and grow in distance from one another as
they get farther from the heated surface. This indicates that the model is in qualitative
agreement with experimental data.
Quantitative experimental data from Ingham [14] was also compared to the
Fluent results to provide model validation.
Figure 3.1.1-6, Figure 3.1.1-7 and
Figure 3.1.1-8 display a comparison of dimensionless temperature versus dimensionless
distance for four dimensionless times at an angle of 30°, 90° and 180°, respectively,
from the positive x-axis. Dimensionless temperature is T = (T’ – T∞) / (Twall – T∞) where
T’ is the actual fluid temperature, T∞ is the bulk fluid temperature and Twall is the heated
wall temperature.
Dimensionless distance is (r’-a)/a * Gr1/4 where r’ is the radial
distance from the heated surface, a is the cylinder diameter and Gr is the Grashof
number.
Dimensionless time is t = t’ * (βgΔT/a)1/2 where t’ is real time, ΔT is
(Twall – T∞), β is the coefficient of thermal expansion and a is the cylinder diameter.
29
(a)
(b)
Figure 3.1.1-6: Dimensionless Temperature at θ = 30°
(a) From Ingham [14] and (b) From Fluent
(a)
(b)
Figure 3.1.1-7: Dimensionless Temperature at θ = 90°
(a) From Ingham [14] and (b) From Fluent
30
(a)
(b)
Figure 3.1.1-8: Dimensionless Temperature at θ = 180°
(a) From Ingham [14] and (b) From Fluent
The heated horizontal cylinder model developed in Fluent showed good
agreement compared with experimental data at the three different radial locations. This
comparison provided confidence that the information obtained from the model was
accurate.
To ensure that the mesh had no significant effect on the results, a mesh validation
was performed.
The mesh validation compared the results shown in this section
(“Mesh 1” in Table 3.1.1-3) to a second mesh with an increased number of finite
volumes (“Mesh 2” in Table 3.1.1-3). The results from the mesh validation displayed in
Table 3.1.1-3 prove that the results are mesh independent.
Table 3.1.1-3: Mesh Validation for Horizontal Cylinder Model
Number of Nodes
Number of Elements
Max Velocity (m/s)
Max Total Temperature (K)
Min Density (kg/m3)
Mesh 1
19716
38688
0.01627
309.9239
993.1765
31
Mesh 2
23636
46400
0.01621
309.9531
993.1625
Difference
19.88 %
19.93 %
-0.37 %
0.01 %
0.00 %
3.1.2
VERTICAL PLATE
Single phase convection heat transfer around a vertical plate with a constant
surface temperature submerged in an infinite pool of liquid at a lower temperature was
also analyzed. Energy passed from the slightly warmer plate to the fluid causing its
temperature to increase and the fluid to rise.
Figure 3.1.2-1 shows a schematic
representation of the geometry and boundary conditions used to model the vertical plate.
The top and bottom walls of the rectangle represent inlet and outlet pressure boundaries
respectively, with pressure conditions set such that the fluid is stagnant until the plate is
heated. The left and right walls of the rectangle are slip boundaries to more accurately
model an infinite pool. See Table 3.1.2-1 for a detailed list of input parameters used.
Figure 3.1.2-1: Vertical Plate Schematic
Figure 3.1.2-2 presents the liquid temperature field after 20 seconds of heating.
When energy is exchanged between the plate and the fluid, a thermal boundary layer is
created. Thermodynamic equilibrium demands that the plate, and the fluid in direct
contact with it, be at the same temperature. The region in which the fluid temperature
changes from the plate surface temperature to that of the bulk fluid temperature is known
as the thermal boundary layer. The teal color in Figure 3.1.2-2 shows the growth of the
thermal boundary layer, which is relatively small at the bottom of the plate but grows
due to heat addition (teal color expands away from the plate) as the fluid climbs.
32
Table 3.1.2-1: Vertical Plate Model Input
Input
Geometry
Plate Height
Plate Width
Pool Height
Pool Width
2D Space
Solver
Time
Time Step Size
Type
Velocity Formulation
Gravity
Models
Energy
Viscous
Density
Initial Conditions
Plate Surface Temperature
Fluid Temperature
Material Properties (Water)
Specific Heat
Thermal Conductivity
Viscosity
Density
Solution Methods
Scheme
Gradient
Pressure
Momentum
Energy
Transient Formulation
Value
0.18 m
0.01 m
0.20 m
0.13 m
Planar
Transient
0.05 s
Pressure Based
Relative
-9.8 m/s2 (Y-direction)
Active
Laminar
Boussinesq
310 K
300 K
4182 J/kg-K
0.6 W/m-K
0.001003 kg/m-s
See Table 3.1.2-2
PISO
Least Square Cell Based
PRESTO!
Second Order Upwind
Second Order Upwind
Second Order Implicit
Table 3.1.2-2: Vertical Plate Model Fluid Density
Density (kg/m3)
999.9
994.1
974.9
958.4
Temperature (K)
273
308
348
373
33
Figure 3.1.2-2: Temperature (K)
Figure 3.1.2-3 shows the liquid velocity in vector form. The figure shows that
the velocity is primarily vertical and the magnitude increases with elevation.
The
increase in fluid velocity with elevation is caused by an increase in energy absorption as
the fluid rises along the heated surface which causes a greater density gradient and
therefore a larger buoyancy force.
Figure 3.1.2-3: Velocity Vectors (m/s)
34
Comparing Figure 3.1.2-3 (vertical plate liquid velocity vectors) with
Figure 3.1.1-4 (horizontal cylinder liquid velocity vectors) produces interesting results.
Because of the larger heated region, it was expected that the vertical plate would produce
a greater maximum fluid velocity compared to the horizontal cylinder. The vertical plate
has a maximum fluid velocity of 0.0149 m/s while the horizontal cylinder has a
maximum fluid velocity of 0.0177 m/s. Although the difference is small, it is notable.
The horizontal cylinder generates a larger maximum velocity because the buoyancy
driving force is not impeded by the drag force created by the heated surface. Although
the vertical plate continues to heat the fluid as it travels upward, the velocity is limited
by friction which causes the plate scenario to have a smaller maximum velocity.
To ensure that the model calculated realistic results, the solution was compared
to experimental data. Figure 3.1.2-4 shows interference fringes surrounding a heated
vertical plate in natural convection. Each interference fringe can be interpreted as a band
of constant density and therefore temperature.
(a)
(b)
Figure 3.1.2-4: Interference Fringes Around a Heated Vertical Plate
(a) From Eckert [13] and (b) Isotherms From Fluent
35
Figure 3.1.2-4 shows that the experimental data and model solution have
isotherms that extend away from the plate and grow in distance from one another as they
get farther from the heated surface. This indicates that the model is in qualitative
agreement with experimental data.
Experimental data from Ostrach [15] was compared to the Fluent results to assess
the quantitative accuracy of the model. Figure 3.1.2-5 and Figure 3.1.2-6 display a
comparison of dimensionless temperature versus dimensionless distance for five
different Prandtl numbers. Figure 3.1.2-5a shows theoretical values and Figure 3.1.2-5b
compares some of the theoretical values to experimental data.
Dimensionless
temperature is H(η) = (T – T∞) / (T0 – T∞) where T is the actual fluid temperature, T∞ is
the bulk fluid temperature and T0 is the wall temperature. Dimensionless distance is
η = (Y / X) * (Grx / 4)1/4 where Grx is the Grashof number, Y is the vertical height and X
is the distance from the plate.
(a)
(b)
Figure 3.1.2-5: Dimensionless Temperature for Various Prandtl Numbers
(a) Theoretical Values and (b) Experimental Values [15]
36
Figure 3.1.2-6: Dimensionless Temperature for Various Prandtl Numbers (Fluent)
The heated vertical plate model developed in Fluent produced results that slightly
overestimate the thickness of the temperature profile when compared to experimental
data for five different Prandtl numbers. The slight over prediction is due to imperfect
extraction of the raw data from Fluent.
To ensure that the mesh had no significant effect on the results, a mesh validation
was performed.
The mesh validation compared the results shown in this section
(“Mesh 1” in Table 3.1.2-3) to a second mesh with an increased number of finite
volumes (“Mesh 2” in Table 3.1.2-3). The results from the mesh validation shown in
Table 3.1.2-3 prove that the results are mesh independent.
Table 3.1.2-3: Mesh Validation for Vertical Plate Model
Number of Nodes
Number of Elements
Max Velocity (m/s)
Max Total Temperature (K)
Min Density (kg/m3)
Mesh 1
12310
23572
0.01376
309.8089
993.2319
37
Mesh 2
18081
35168
0.01380
309.7991
993.2365
Difference
46.88 %
49.19 %
0.29 %
0.00 %
0.00 %
3.2
LAMINAR FLOW
A steady state, axisymmetric, laminar flow model was developed. Figure 3.2-1
shows a schematic representation of the geometry and boundary conditions used to
model laminar flow within a pipe. The bottom line of the rectangle is an axis of rotation
which is used to simplify the geometry and represents the pipe centerline. The top line
of the rectangle is a no slip boundary and after the rotation, becomes the pipe wall. The
left and right lines of the rectangle are the inlet and outlet areas respectively, which
when revolved, are circular. See Table 3.2-1 for a detailed list of input parameters used.
Figure 3.2-1: Laminar Flow Schematic
Based on the selected inlet conditions, the Reynolds number is 352, which is well
within the laminar regime. Figure 3.2-2 displays the velocity magnitude versus position
(distance from the pipe centerline) at different lengths from the pipe entrance. For
example, “line-10cm” is the velocity profile 10 cm from the pipe entrance.
The
parabolic shape of the velocity profile is clearly visible which is characteristics of
laminar flow.
Figure 3.2-2: Velocity Magnitude
38
Fluid velocity within the pipe slowly decreases as distance from the pipe centerline
increases. Also, as the flow develops, the entrance effects dissipate, the velocity profile
becomes more parabolic until it reaches a steady state at about 45 cm from the entrance
which is in good agreement with well known “entrance length” calculations [5].
Table 3.2-1: Laminar Flow Model Input
Input
Value
Geometry
Pipe Diameter
Pipe Length
2D Space
Solver
Time
Type
Velocity Formulation
Gravity
Models
Energy
Viscous
Material Properties (Water)
Specific Heat
Thermal Conductivity
Viscosity
Density
Inlet Conditions
Pipe Wall Surface Temperature
Fluid Temperature
Fluid Velocity
Solution Methods
Scheme
Gradient
Pressure
Momentum
Energy
0.03 m
0.50 m
Axisymmetric
Steady
Pressure Based
Relative
-9.8 m/s2 (X-direction)
Active
Laminar
4182 J/kg-K
0.6 W/m-K
0.001003 kg/m-s
See Table 3.2-2
305 K
300 K
0.05 m/s
Coupled
Least Square Cell Based
Second Order
Second Order Upwind
Second Order Upwind
Table 3.2-2: Laminar Flow Model Fluid Density
Density (kg/m3)
999.9
994.1
Temperature (K)
273
308
39
Another characteristic of laminar flow is the lack of mixing that occurs within the
fluid. The radial velocity within the pipe is basically zero and each fluid element
remains about the same distance from the centerline from entrance to exit. Figure 3.2-3
displays the radial flow velocity. As expected, the radial velocity for most of the pipe is
near zero and is less than 10-3 times the average axial velocity. Radial velocity is at a
maximum near the entrance of the pipe due to inlet boundary conditions and entrance
effects but these have a negligible impact on system as a whole.
Figure 3.2-3: Radial Velocity (m/s)
Figure 3.2-4 provides the temperature profile for the laminar flow analyzed.
Because there is little to no radial velocity, convection and conduction are the primary
forms of heat transfer which causes the thermal boundary layer to grow at a very slow
rate.
The growth of the thermal boundary layer is shown in Figure 3.2-4 by the
expansion of the teal colored region.
Figure 3.2-4: Temperature (K)
Figure 3.2-5 shows the wall shear stress as a function of distance from the pipe
entrance. The wall stress is much larger in the first 10 cm due to entrance effects. Once
the entrance effects dissipate, the wall shear stress slowly decreases as the flow reaches a
steady state.
40
Figure 3.2-5: Wall Shear Stress
To ensure that the mesh had no significant effect on the results, a mesh validation
was performed.
The mesh validation compared the results shown in this section
(“Mesh 1” in Table 3.2-3) to a second mesh with an increased number of finite volumes
(“Mesh 2” in Table 3.2-3).
The results from the mesh validation displayed in
Table 3.2-3 prove that the results are mesh independent.
Table 3.2-3: Mesh Validation for Laminar Flow Model
Number of Nodes
Number of Elements
Max Velocity (m/s)
Min Radial Velocity (m/s)
Max Dynamic Pressure (Pa)
Max Temperature (K)
Mesh 1
26320
25353
0.079561
-0.003293
3.15925
304.6503
41
Mesh 2
31000
29970
0.079507
-0.003528
3.155022
304.6855
Difference
17.78 %
18.21 %
-0.07 %
7.12 %
-0.13 %
0.01 %
3.3
TURBULENT FLOW
3.3.1
TURBULENT FLOW WITHOUT HEAT TRANSFER
A steady state, axisymmetric, turbulent flow model was developed.
Figure 3.3.1-1 shows a schematic representation of the geometry and boundary
conditions used to model turbulent flow within a pipe without heat transfer. The bottom
line of the rectangle is an axis of rotation which is used to simplify the geometry and
represents the pipe centerline. The top line of the rectangle is a no slip boundary and
after the rotation becomes the pipe wall. The left and right lines of the rectangle are the
inlet and outlet areas respectively, which when revolved, are circular. See Table 3.3.1-1
for a detailed list of input parameters used.
Figure 3.3.1-1: Turbulent Flow Without Heat Transfer Schematic
Based on the selected inlet conditions, the Reynolds number is 42314, which is
well within the turbulent regime. Figure 3.3.1-2 displays the velocity magnitude versus
position (distance from the pipe centerline) at different distances from the pipe entrance.
Figure 3.3.1-2: Velocity Magnitude
42
The velocity profile of turbulent flow differs significantly in two ways compared
to the velocity profile of laminar flow (Figure 3.2-2). First, turbulent flow velocity
profiles are much flatter. Therefore, the fluid velocity doesn’t decrease significantly
until close to the pipe wall. Second, entrance effects dissipate much quicker in turbulent
flow [5] and thus the fluid velocity reaches a steady state velocity profile in a shorter
distance. Figure 3.3.1-2 (turbulent flow) shows that flow reaches a steady profile about
10 cm from the pipe entrance. Figure 3.2-2 (laminar flow) shows that flow reaches a
steady profile about 45 cm from the pipe entrance.
This qualitatively matches
experimental data well.
Table 3.3.1-1: Turbulent Flow Without Heat Transfer Model Input
Input
Value
Geometry
Pipe Diameter
0.03 m
Pipe Length
0.50 m
2D Space
Axisymmetric
Solver
Time
Steady
Type
Pressure Based
Velocity Formulation
Relative
Gravity
-9.8 m/s2 (X-direction)
Models
Energy
Inactive
Viscous
Realizable k-ε
Turbulence Model
Near Wall Treatment
Enhanced
Turbulent Intensity
0.0422483 *
Inlet Conditions
Fluid Mass Flow Rate
1.0 kg/s
Material Properties (Water)
Density
998.2 kg/m3
Viscosity
0.001003 kg/m-s
Solution Methods
Scheme
Coupled
Gradient
Least Square Cell Based
Pressure
Second Order
Momentum
Second Order Upwind
Turbulent Kinetic Energy
Second Order Upwind
Turbulent Dissipation Rate
Second Order Upwind
* Calculation shown in Table 2.5.1-2.
43
Figure 3.3.1-3 displays the wall shear stress versus distance from the pipe
entrance.
The shear stress is very large at the pipe entrance and decays to the steady
state value after about 10 cm (same location where the velocity profile reaches steady
state). The large increase in shear stress at the beginning of the pipe (~1-2 cm from the
inlet) is caused by entrance effects.
Figure 3.3.1-4 shows that that maximum radial
velocity occurs near the pipe entrance. Figure 3.3.1-5 reveals that the greatest reduction
in axial velocity occurs near the pipe entrance which is necessary to conserve
momentum when radial velocity increases. Since shear stress is related to change in
velocity parallel to the wall (axial velocity), the increase in wall shear stress near the
pipe entrance is reasonable.
Figure 3.3.1-3: Wall Shear Stress
Figure 3.3.1-4: Radial Velocity (m/s)
Figure 3.3.1-5: 𝒅𝒖𝒙 ⁄𝒅𝒙
44
To further investigate the impact of entrance effects, two additional scenarios
were examined using a mass flow rate of 0.5 kg/s (Figure 3.3.1-6) and a mass flow rate
of 1.5 kg/s (Figure 3.3.1-7).
(a)
(b)
(c)
Figure 3.3.1-6: Results for a Mass Flow Rate of 0.5 kg/s
(a) Radial Velocity (m/s) (b) Wall Shear Stress (c) 𝒅𝒖𝒙 ⁄𝒅𝒙
(a)
(b)
(c)
Figure 3.3.1-7: Results for a Mass Flow Rate of 1.5 kg/s
(a) Radial Velocity (m/s) (b) Wall Shear Stress (c) 𝒅𝒖𝒙 ⁄𝒅𝒙
45
Figures 3.3.1-6 and 3.3.1-7 prove that the maximum wall shear stress and the
maximum radial velocity are directly related to mass flow rate. At a certain distance
from the pipe entrance, the change in axial velocity as a function of position reaches zero
and the wall shear stress reaches a constant value. The pipe length necessary to reach a
steady state shear stress is also related to the mass flow rate. A larger mass flow rate
requires a greater distance to reach a constant shear stress.
Figure 3.3.1-8 shows that most of the turbulent kinetic energy is located near the
pipe wall due to shear stress.
Figure 3.3.1-8: Turbulent Kinetic Energy (m2/s2)
Figure 3.3.1-9 shows the production of turbulent kinetic energy as a function of
distance. The trend of Figure 3.3.1-9 is similar to that of Figure 3.3.1-3 because shear
stress, created by the wall, produces turbulent kinetic energy.
Figure 3.3.1-9: Production of Turbulent Kinetic Energy
A mesh validation was not performed for this model directly.
The mesh
accuracy is proven adequate in Section 3.3.2 which utilizes the same model with the
addition of energy transfer from the pipe wall to the fluid.
46
3.3.2
TURBULENT FLOW WITH HEAT TRANSFER
The turbulent flow model described in Section 3.3.1 was enhanced to include
heat transfer from the pipe wall to the fluid.
Figure 3.3.2-1 shows a schematic
representation of the geometry and boundary conditions used to model turbulent flow in
a pipe with heat transfer. The bottom line of the rectangle is an axis of rotation which is
used to simplify the geometry and represents the pipe centerline. The top line of the
rectangle is a no slip boundary with a constant heat flux and after the rotation becomes
the pipe wall. The left and right lines of the rectangle are the inlet and outlet areas
respectively, which when revolved, are circular. See Table 3.3.2-1 for a detailed list of
input parameters used.
Figure 3.3.2-1: Turbulent Flow With Heat Transfer Schematic
Figure 3.3.2-2 displays the fluid temperature change caused by energy addition
from the pipe wall. The radial temperature distribution in Figure 3.3.2-2 is more evenly
distributed than the radial temperature distribution in Figure 3.2-4 (laminar flow).
Uniform temperature distribution is a characteristic of turbulent flow and made possible
by the chaotic nature of the flow regime.
Figure 3.3.2-2: Temperature (K)
The radial velocity in Figure 3.3.2-3 is very similar to that in Figure 3.3.1-4
which means that the heat addition has a negligible impact on fluid velocity. If the heat
transfer rate to the fluid was increased sufficiently such that flow velocity was impacted,
then the radial velocity between the two scenarios would also differ.
47
Table 3.3.2-1: Turbulent Flow With Heat Transfer Model Input
Input
Value
Geometry
Pipe Diameter
0.03 m
Pipe Length
0.50 m
2D Space
Axisymmetric
Solver
Time
Steady
Type
Pressure Based
Velocity Formulation
Relative
Gravity
-9.8 m/s2 (X-direction)
Models
Energy
Active
Viscous
Realizable k-ε
Turbulence Model
Near Wall Treatment
Enhanced
Turbulent Intensity
0.0422483 *
Inlet Conditions
Fluid Mass Flow Rate
1.0 kg/s
Fluid Temperature
300 K
Wall Heat Flux
450 kW/m2
Material Properties (Water)
Specific Heat
4182 J/kg-K
Thermal Conductivity
0.6 W/m-K
Viscosity
0.001003 kg/m-s
Density
See Table 3.3.2-2
Solution Methods
Scheme
Coupled
Gradient
Least Square Cell Based
Pressure
Second Order
Momentum
Second Order Upwind
Turbulent Kinetic Energy
Second Order Upwind
Turbulent Dissipation Rate
Second Order Upwind
Energy
Second Order Upwind
* Calculation shown in Table 2.5.1-2.
Table 3.3.2-2: Turbulent Flow With Heat Transfer Model Fluid Density
Density (kg/m3)
999.9
994.1
974.9
Temperature (K)
273
308
348
48
Figure 3.3.2-3: Radial Velocity (m/s)
Closely comparing the velocity profiles for the two turbulent flow models
(Figure 3.3.1-2 and Figure 3.3.2-4) reveals that the velocity magnitude is slightly larger
for the case with heat transfer. The energy addition causes the density of the fluid to
decrease and the velocity increases slightly to maintain a constant mass flow through the
pipe.
Figure 3.3.2-4: Velocity Magnitude
As expected, because the velocity magnitudes are similar, the wall shear stress
shown in Figure 3.3.2-5 matches the wall shear stress shown in Figure 3.3.1-3.
Figure 3.3.2-5: Wall Shear Stress
49
Comparing the velocity magnitude, radial velocity and wall shear stress from
Section 3.3.1 to Section 3.3.2 proves that the addition of heat transfer in this case has a
negligible impact on the turbulent flow.
This is reasonable since the heat flux is
relatively small and does not create any localized phase change. Thus, the relationships
developed in Section 3.3.1 (impact mass flow has on shear stress and radial velocity) are
applicable to turbulent flows with heat transfer as long as the heat transfer rate is small.
To ensure that the mesh had no significant effect on the results, a mesh validation
was performed.
The mesh validation compared the results shown in this section
(“Mesh 1” in Table 3.3.2-3) to a second mesh with an increased number of finite
volumes (“Mesh 2” in Table 3.3.2-3). The results from the mesh validation shown in
Table 3.3.2-3 prove that the results are mesh independent.
Table 3.3.2-3: Mesh Validation for Turbulent Flow With Heat Transfer Model
Number of Nodes
Number of Elements
Max Velocity (m/s)
Max Temperature (K)
Min Density (kg/m3)
Max Dynamic Pressure (Pa)
3.4
TWO-PHASE FLOW
3.4.1
GAS MIXING TANK
Mesh 1
31031
31000
1.502045
317.6659
989.4604
1122.853
Mesh 2
35739
34624
1.500343
318.1447
989.2305
1119.909
Difference
15.17 %
11.69 %
-0.11 %
0.15 %
-0.02 %
-0.26 %
In many branches of engineering, gas injection techniques have been extensively
utilized to enhance chemical reaction rates, homogenize temperature and chemical
compositions, and remove impurities. In the steel industry, the advancements made in
mixing have increased the level of control over the steelmaking process which has
improved the quality of steel produced. To mix the molten metal, gas is pumped through
a porous plug located at the bottom of the mixing tank. The porous plug controls the
velocity and bubble diameter of the gas. Buoyancy forces cause the injected gas to
move quickly through the molten metal and drag forces causes mixing.
50
A transient, 2D Cartesian, gas mixing tank model was developed using the
Eulerian multiphase model. Figure 3.4.1-1 shows a schematic representation of the
geometry and boundary conditions used to model the gas mixing tank. The top line of
the rectangle is a pressure outlet and the left, right and most of the bottom lines of the
rectangle represent no slip boundaries. The red line on the bottom of the rectangle
represents a velocity inlet and is where the gas jet enters the tank to mix the liquid. See
Table 3.4.1-1 for a detailed list of input parameters used.
Figure 3.4.1-1: Gas Mixing Tank Schematic
Figure 3.4.1-2 and Figure 3.4.1-3 show the computed gas volume fraction and
Figure 3.4.1-4 and Figure 3.4.1-4 show the liquid vector velocity and the gas vector
velocity, respectively, after 5 seconds of gas injection. Midway through the liquid
volume in Figure 3.4.1-2, the gas jet begins to become wavy. The wavy behavior is
explained by Rayleigh instability which states that streams tend to breakup into smaller
packets with the same volume but less surface area due to perturbations that grow over
time. The length required for the jet to breakup is dependent upon the jet radius and
surface tension. Figure 3.4.1-3 shows the gas volume fraction at the jet centerline from
the bottom of the tank (0.00 m) to the top of the tank (0.50 m). The gas volume fraction
oscillates between 1.0 (gas) and 0.1 (mostly liquid) which indicates that the jet is
breaking into discrete bubbles. This further demonstrates the effect of jet breakup due to
Rayleigh instability.
51
Table 3.4.1-1: Gas Mixing Tank Model Input
Input
Geometry
Tank Width
Tank Height
Porous Plug Width
2D Space
Solver
Time
Time Step Size
Type
Velocity Formulation
Gravity
Models
Energy
Viscous
Multiphase
Drag
Turbulence Model
Near Wall Treatment
Turbulent Kinetic Energy
Turbulent Dissipation Rate
Initial Conditions
Water Level
Gas Velocity
Bubble Diameter
Material Properties (Water)
Density
Viscosity
Material Properties (Air)
Density
Viscosity
Surface Tension
Solution Methods
Scheme
Gradient
Momentum
Volume Fraction
Turbulent Kinetic Energy
Turbulent Dissipation Rate
Transient Formulation
52
Value
0.30 m
0.60 m
0.02 m
Planar
Transient
0.001 s
Pressure Based
Relative
-9.8 m/s2 (Y-direction)
Inactive
Standard k-ε
Eulerian
Schiller-Nauman
Standard
0 m2/s2
0 m2/s3
0.40 m
0.5 m/s
0.001 m
998.2 kg/m3
0.001003 kg/m-s
1.225 kg/m3
1.7894E-05 kg/m-s
0.072 N/m
Phase Coupled SIMPLE
Least Square Cell Based
Second Order Upwind
QUICK
Second Order Upwind
Second Order Upwind
Second Order Implicit
Figure 3.4.1-2: Gas Volume Fraction
Figure 3.4.1-3: Gas Volume Fraction at Jet Centerline
53
The liquid and gas velocities displayed in Figure 3.4.1-4 and Figure 3.4.1-5,
respectively, are similar in trend and magnitude which indicates that the drag force
between the two phases is strong (slip ratio close to one). The maximum gas velocity
(1.593 m/s) is much greater than the inlet gas velocity (0.5 m/s); therefore, buoyancy
forces are significant. Figure 3.4.1-4 shows that there is a number of small eddies
created by the injected gas which provide a significant amount of mixing within the
liquid. These eddies are responsible for the even distribution of alloying elements
during the steel making process.
Figure 3.4.1-4: Liquid Velocity Vectors (m/s)
54
Figure 3.4.1-5: Gas Velocity Vectors (m/s)
To ensure that the mesh had no significant effect on the results, a mesh validation
was performed. The mesh validation compared the results displayed in this section
(“Mesh 1” in Table 3.4.1-2) to a second mesh with an increased number of finite
volumes (“Mesh 2” in Table 3.4.1-2). The results from the mesh validation shown in
Table 3.4.1-2 prove that the results are mesh independent.
Table 3.4.1-2: Mesh Validation for Gas Mixing Tank Model
Number of Nodes
Number of Elements
Max Liquid Velocity (m/s)
Max Gas Velocity (m/s)
Max Static Pressure (Pa)
Max Liquid Total Pressure (Pa)
Max Liquid Volume Fraction
Mesh 1
30625
30256
1.539086
2.046923
3925.424
4775.512
1.000000
55
Mesh 2
36045
35644
1.453488
2.086285
3894.616
4732.633
1.000000
Difference
17.70 %
17.81 %
-5.56 %
1.92 %
-0.78 %
-0.90 %
0.00 %
3.4.2
BUBBLE COLUMN
A bubble column reactor is a tool primarily used to study gas-liquid reactions.
The apparatus is a vertical column of liquid with gas introduced continuously at the
bottom through a sparger. The bubble column contains gas dispersed as bubbles in a
continuous volume of liquid. Per Section 2.6, the flow is considered to be bubbly. The
gas introduced through the sparger provides mixing, similar to the gas mixing tank in
Section 3.4.1 but much less intense. This method of mixing is less invasive and requires
less energy than mechanical stirring. Bubble column reactors are often used in industry
to develop and produce chemicals and fuels for use in chemical, biotechnology, and
pharmaceutical processes.
A transient, 2D Cartesian, bubble column model was developed using the
Eulerian multiphase model. Figure 3.4.2-1 shows a schematic representation of the
geometry and boundary conditions used to model the bubble column. The top line of the
rectangle is a pressure outlet and the left and right lines of the rectangle represent no slip
boundaries. The bottom line of the rectangle signifies a velocity inlet and is where the
gas bubbles enter the column. See Table 3.4.2-1 for a detailed list of input parameters
used.
Figure 3.4.2-1: Bubble Column Schematic
56
Table 3.4.2-1: Bubble Column Model Input
Input
Geometry
Column Width
Column Height
2D Space
Solver
Time
Time Step Size
Type
Velocity Formulation
Gravity
Models
Energy
Viscous
Multiphase
Drag
Turbulence Model
Near Wall Treatment
Turbulent Kinetic Energy
Turbulent Dissipation Rate
Initial Conditions
Water Level
Gas Flow Rate
Bubble Diameter
Material Properties (Water)
Density
Viscosity
Material Properties (Air)
Density
Viscosity
Surface Tension
Solution Methods
Scheme
Gradient
Momentum
Volume Fraction
Turbulent Kinetic Energy
Turbulent Dissipation Rate
Transient Formulation
57
Value
0.10 m
0.75 m
Planar
Transient
0.001 s
Pressure Based
Relative
-9.8 m/s2 (Y-direction)
Inactive
Standard k-ε
Eulerian
Schiller-Nauman
Standard
0 m2/s2
0 m2/s3
0.50 m
0.05 m/s
0.005 m
998.2 kg/m3
0.001003 kg/m-s
1.225 kg/m3
1.7894E-05 kg/m-s
0.072 N/m
Phase Coupled SIMPLE
Least Square Cell Based
Second Order Upwind
QUICK
Second Order Upwind
Second Order Upwind
Second Order Implicit
Figure 3.4.2-2 is a comparison between the gas volume fraction 1 second and
5 seconds after gas has begun flowing through the bubble column. After 5 seconds, the
gas reaches the top of the liquid and causes the surface to change shape. Compared to
the initial liquid level, the level after 5 seconds is about 5 cm higher. The level increase
is known as gas holdup and is caused by phase drag forces and displacement.
Figure 3.4.2-2b reveals that most of the gas travels along the wall in a quasi-annular
fashion known as wall-peaking bubbly flow.
(a)
(b)
Figure 3.4.2-2: Gas Volume Fraction
After (a) 1 Second and (b) 5 Seconds
Figure 3.4.2-3 is a comparison between the liquid velocity vectors 1 second and
5 seconds after the gas has begun flowing through the bubble column. Distinct paths of
liquid movement, primarily along the walls, can be seen at both time points. Due to
buoyancy and phase drag forces, the largest liquid velocities coincide with the regions of
greatest gas volume fraction.
58
(a)
(b)
Figure 3.4.2-3: Liquid Velocity Vectors (m/s)
After (a) 1 Second and (b) 5 Seconds
Figure 3.4.2-4 is a comparison between the gas velocity vectors 1 second and
5 seconds after gas has begun flowing through the bubble column. The white region
two-thirds up the bubble column in Figure 3.4.2-4a is a region where the gas has not
reached. It is noteworthy that the original gas-liquid interface is not flat but consists of
two parabolas. The two parabolas were created because most of the gas travels close to
the wall. Figure 3.4.2-4b reveals that the greatest gas velocities occur near the walls
which are also the areas of greatest gas volume fraction. Higher gas volume fractions
lead to greater buoyancy forces which cause greater gas velocities.
A second scenario was analyzed to compare the impact that gas inlet velocity has
on gas holdup. This case is the same as the case described in Table 3.4.2-1 except that
the gas inlet velocity was increased to 0.10 m/s. Figure 3.4.2-5 illustrates the gas
volume fraction 1 second and 5 seconds after the gas has begun flowing through the
bubble column. Figure 3.4.2-5b reveals that the injected gas causes the water level to
rise about 15 cm. This is a much larger increase than the level increase shown in
Figure 3.4.2-2b, which employs a gas inlet velocity of 0.05 m/s.
59
(a)
(b)
Figure 3.4.2-4: Gas Velocity Vectors (m/s)
After (a) 1 Second and (b) 5 Seconds
(a)
(b)
Figure 3.4.2-5: Gas Volume Fraction (0.10 m/s)
After (a) 1 Second and (b) 5 Seconds
60
To ensure that the mesh had no significant effect on the results, a mesh validation
was performed. The mesh validation compared the results from the scenario with a gas
velocity of 0.05 m/s (“Mesh 1” in Table 3.4.2-2) to a second mesh with an increased
number of finite volumes (“Mesh 2” in Table 3.4.2-2). The results from the mesh
validation displayed in Table 3.4.2-2 prove that the results are mesh independent.
Table 3.4.2-2: Mesh Validation for Bubble Column Model
Number of Nodes
Number of Elements
Max Liquid Velocity (m/s)
Average Gas Velocity (m/s)
Max Static Pressure (Pa)
Max Liquid Volume Fraction
3.4.3
Mesh 1
7006
6750
0.625945
0.313947
4929.094
0.998733
Mesh 2
8785
8500
0.63157
0.308535
4920.58
1.00000
Difference
25.39 %
25.93 %
0.90 %
1.72 %
-0.17 %
0.13 %
BUBBLE COLUMN WITH POPULATION BALANCE MODEL
The bubble column model created in Section 3.4.2 was expanded to include a
population balance model (PBM) with three discrete bubble sizes so that bubble swarm
could be tracked. In all gas-liquid flows, the bubbles can increase or decrease in size
due to coalescence or breakup. Coalescence occurs when two or more bubbles collide
and the liquid barrier between them ruptures to form a larger bubble. Bubble breakup
occurs when a bubble collides with a turbulent eddy approximately equal to its size
causing it to split into two or more smaller bubbles. Table 3.4.3-1 lists the input used to
create the population balance model implemented.
Figure 3.4.3-1 is a comparison between the gas volume fraction at 1 second and
5 seconds after gas has begun flowing through the bubble column. When comparing
Figure 3.4.3-1 to Figure 3.4.2-2, there are noticeable differences. One of the obvious
differences between the two figures is the distribution of the gas phase at the two time
points. With the population balance model implemented (Figure 3.4.3-1), the gas phase
distribution is more uniform and does not contain any areas with large gas volume
fractions. This is most noticeable at the bottom of the bubble column after 5 seconds.
61
Table 3.4.3-1: Population Balance Model Input
Input
Method
Number of Bins
Bin-0
Bin-1
Bin-2
Bin Distribution
Bin-0
Bin-1
Bin-2
Aggregation Kernel
Model
Surface Tension
Breakage Kernel
Model
Surface Tension
Formulation
Value
Discrete
3
0.0075595 m
0.0047622 m
0.0030000 m
25 %
50 %
25 %
Luo
0.072 N/m
Luo
0.072 N/m
Hagesather
(a)
(b)
Figure 3.4.3-1: Gas Volume Fraction with PBM
After (a) 1 Second and (b) 5 Seconds
62
Figure 3.4.3-2 is a comparison between the liquid velocity vectors at 1 second
and 5 seconds after gas has begun flowing through the bubble column. Figure 3.4.3-2b
reveals that the liquid velocity increases as elevation increases. This is less noticeable in
Figure 3.4.2-3 which displays a more uniform liquid velocity. Table 3.4.3-2 shows that
there is a greater number of large bubbles at the outlet compared to the inlet in
Figure 3.4.3-2b. The larger bubbles attain higher velocities due to greater buoyancy
forces which in turn increases the liquid velocity due to drag between the two phases.
The velocity gradient in Figure 3.4.2-3 is more uniform because the bubbles do not
coalesce and therefore maintain a constant buoyancy force.
(a)
(b)
Figure 3.4.3-2: Liquid Velocity Vectors with PBM (m/s)
After (a) 1 Second and (b) 5 Seconds
The population balance model calculates the bubble size distribution at each axial
height using the Luo breakup and coalescence model. Table 3.4.3-2 lists the bubble size
distribution at the inlet and outlet of the bubble column. This table proves that there is a
strong bias for the smaller bubbles to coalesce into larger bubbles; thus, surface tension
63
is a strong driver to reduce total surface area. This also proves that there is very little
turbulence within the column to cause the bubbles to break apart.
Table 3.4.3-2: Bubble Size Distribution – Surface Tension of 0.072 N/m
Inlet
(Fraction)
0.250
0.500
0.250
Bin-0 (0.0076 m)
Bin-1 (0.0048 m)
Bin-2 (0.0030 m)
Outlet
(Fraction)
0.865
0.117
0.018
Net
(Fraction)
+0.615
-0.383
-0.232
Figure 3.4.3-3 is a comparison between the gas velocity vectors at 1 second and
5 seconds after gas has begun flowing through the bubble column.
Similar to
Figure 3.4.2-4, the shape of the gas as it initially climbs the bubble column is made up of
two adjacent parabolas; however, it is much more severe in Figure 3.4.3-3a.
Figure 3.4.3-3b reveals that the gas velocity increases as elevation increases due to
bubble coalescence.
(a)
(b)
Figure 3.4.3-3: Gas Velocity Vectors with PBM (m/s)
After (a) 1 Second and (b) 5 Seconds
64
The impact that surface tension has on bubble size distribution was evaluated by
reducing it by a factor of ten to 0.0072 N/m. Table 3.4.3-3 displays the bubble size
distribution at the inlet and outlet of the bubble column with the reduced surface tension.
The smaller surface tension decreases the driving force for bubbles to coalesce and
significantly reduces the average bubble diameter.
Table 3.4.3-3: Bubble Size Distribution – Surface Tension of 0.0072 N/m
Bin-0 (0.0076 m)
Bin-1 (0.0048 m)
Bin-2 (0.0030 m)
Inlet
(Fraction)
0.250
0.500
0.250
Outlet
(Fraction)
0.495
0.335
0.170
Net
(Fraction)
+0.245
-0.165
-0.080
A mesh validation was not performed for this model directly. The mesh quality
is proven adequate in Section 3.4.2 which utilizes the same model without the population
balance model employed.
3.5
BOILING HEAT TRANSFER
3.5.1
POOL BOILING
Pool boiling occurs when a liquid transforms to a vapor due to energy absorption
in a fluid that is stagnant. When the temperature of a heated surface sufficiently exceeds
the saturation temperature of the liquid in direct contact with it, vapor bubbles nucleate
on the surface. The bubbles grow until they detach from the surface and move out into
the bulk liquid. While rising is the result of buoyancy, the bubbles either collapse or
continue to grow depending upon whether the liquid is locally subcooled or superheated.
Pool boiling involves complex fluid motions initiated and maintained by the nucleation,
growth, departure and collapse of bubbles, and by natural convection. [11]
A transient, 2D Cartesian, pool boiling model was developed using the Eulerian
multiphase model. Figure 3.5.1-1 shows a schematic representation of the geometry and
boundary conditions used to model pool boiling. The top line of the rectangle is a
pressure outlet and the bottom line of the rectangle is the heated surface. The left and
65
right lines of the rectangle represent no slip boundaries. See Table 3.5.1-1 for a detailed
list of input parameters used.
Figure 3.5.1-1: Pool Boiling Schematic
Table 3.5.1-1: Pool Boiling Model Input
Input
Geometry
Column Width
Column Height
2D Space
Solver
Time
Time Step Size
Type
Velocity Formulation
Gravity
Models
Energy
Viscous
Multiphase
Drag
Slip
Mass Transfer
Initial Conditions
Bubble Diameter
Initial Fluid Temperature
Heater Temperature (Bottom)
66
Value
0.01 m
0.05 m
Planar
Transient
0.002 s
Pressure Based
Relative
-9.8 m/s2 (Y-direction)
Active
Laminar
Mixture
Schiller-Nauman
Manninen et al.
Evaporation-Condensation
0.0002 m
372 K
383 K
Backflow Temperature (Top)
Backflow Volume Fraction (Top)
Material Properties (Water) [16]
Density
Specific Heat
Thermal Conductivity
Viscosity
Heat of Vaporization
Material Properties (Vapor) [16]
Density
Specific Heat
Viscosity
Thermal Conductivity
Surface Tension
Solution Methods
Scheme
Gradient
Pressure
Momentum
Volume Fraction
Energy
Transient Formulation
373 K
0
See Table 3.5.1-2
4182 J/kg-K
0.6 W/m-K
0.001003 kg/m-s
2.418379E+08 J/kgmol
0.5542 kg/m3
2014 J/kg-K
1.34E-05 kg/m-s
0.0261 W/m-K
0.072 N/m
PISO
Least Square Cell Based
Body Force Weighted
Second Order Upwind
QUICK
Second Order Upwind
Second Order Implicit
Table 3.5.1-2: Pool Boiling Model Fluid Density
Density (kg/m3)
974.9
958.4
Temperature (K)
348
373.15
Figure 3.5.1-2 displays the instantaneous gas volume fraction after 0.9 seconds
and 1.7 seconds of heating. The first time point was chosen because it shows vapor
releasing from the heated surface and entering the bulk fluid which is the driving force
behind most of the fluid motion. The second time point was chosen because it reveals
the interaction between the liquid and vapor at a high level.
The evolution of vapor generation, upward movement (due to buoyancy) and
liquid refill is illustrated in Figures 3.5.1-2 through 3.5.1-4. Figure 3.5.1-2a reveals that
the bottom of the control volume is heated and some vapor has formed (two areas of
significant vapor generation are green). Figure 3.5.1-2b shows that the vapor has moved
67
upward (teal region) and that liquid has moved downward to take its place (blue area at
the bottom).
(a)
(b)
Figure 3.5.1-2: Vapor Volume Fraction
After (a) 0.9 Seconds and (b) 1.7 Seconds
Figure 3.5.1-3 and Figure 3.5.1-4 display the liquid and gas velocities,
respectively, at the two time points. Comparing these two figures indicates that the
largest upward liquid and vapor velocities occur in generally the same regions. These
regions also coincide with the areas of largest gas volume fraction (Figure 3.5.1-2). As
vapor bubbles form on the heated surface, they eventually detach and enter the liquid
above. Due to buoyancy, the vapor travels upward through the liquid. Drag forces
between the two phases cause the liquid to also travel upwards but at a slower rate due to
slip. Other areas of high liquid velocity occur between the two swells of upward moving
vapor and along the walls. The liquid being of greater density flows downward to refill
the void created by the recently generated vapor.
68
(a)
(b)
Figure 3.5.1-3: Liquid Velocity Vectors (m/s)
After (a) 0.9 Seconds and (b) 1.7 Seconds
(a)
(b)
Figure 3.5.1-4: Vapor Velocity Vectors (m/s)
After (a) 0.9 Seconds and (b) 1.7 Seconds
69
Figure 3.5.1-5 shows the volume fraction of vapor on the heated surface after
two seconds. This figure illustrates that vapor is produced significantly at two locations
(vapor volume fraction is at a maximum), 0.0008 m and 0.0095 m.
In this situation,
0.00 m is the left wall and 0.01 m is the right wall. The vapor volume fraction is at a
minimum at approximately 0.005 m which is the location where liquid is replacing the
recently created vapor.
Figure 3.5.1-5: Volume Fraction of Vapor on Heated Surface
To ensure that the mesh had no significant effect on the results, a mesh validation
was performed.
The mesh validation compared the results shown in this section
(“Mesh 1” in Table 3.5.1-3) to a second mesh with an increased number of finite
volumes (“Mesh 2” in Table 3.5.1-3). The results from the mesh validation displayed in
Table 3.5.1-3 prove that the results are mesh independent.
Table 3.5.1-3: Mesh Validation for Pool Boiling Model
Number of Nodes
Number of Elements
Min Mixture Density (kg/m3)
Max Mixture Velocity (m/s)
Max Static Pressure (Pa)
Max Phase Transfer (kg/m3-s)
Min Liquid Volume Fraction
Mesh 1
26645
26208
754.389
0.059396
452.2354
2.169675
0.787011
70
Mesh 2
32481
32000
742.115
0.062788
452.2388
2.190905
0.774197
Difference
21.90 %
22.10 %
-1.63 %
5.71 %
0.00 %
0.98 %
-1.63 %
3.5.2
SUBCOOLED FLOW BOILING
A steady state, axisymmetric, subcooled flow boiling model was developed using
the Eulerian multiphase model. Figure 3.5.2-1 shows a schematic representation of the
geometry and boundary conditions used to model subcooled flow boiling. The bottom
line of the rectangle is an axis of rotation which is used to simplify the geometry and
represents the pipe centerline. The top line of the rectangle is a no slip boundary with a
constant heat flux and after the rotation becomes the pipe wall. The left and right lines
of the rectangle are the inlet and outlet areas respectively, which when revolved, are
circular. In this scenario, the fluid flows in the axial (x) direction and against gravity.
See Table 3.5.2-1 for a detailed list of input parameters used.
Figure 3.5.2-1: Subcooled Flow Boiling Model Schematic
Table 3.5.2-1: Subcooled Flow Boiling Model Input
Input
Value
Geometry
Pipe Diameter
Pipe Length
2D Space
Solver
Time
Type
Velocity Formulation
Gravity
Models
Energy
Viscous
Near Wall Treatment
Turbulent Intensity
Multiphase
Drag
Lift
Heat Transfer
Mass Transfer
Bubble Departure Diameter
Nucleation Site Density
71
0.03 m
0.50 m
Axisymmetric
Steady
Pressure Based
Relative
-9.8 m/s2 (X-direction)
Active
Realizable k-ϵ
Enhanced
0.042079 *
Eulerian
Schiller-Nauman
Boiling-Moraga
Ranz-Marshall
RPI Boiling
Tolubinski-Kostanchuk
Lemmert-Chawla
Bubble Departure Frequency
Cole
Interfacial Area
Ia-Symmetric
Bubble Diameter
Sauter-Mean
Inlet Conditions
Mass Flow Rate
0.3 kg/s
Inlet Fluid Temperature
370 K
Wall Heat Flux
90000 W/m2
Material Properties (Water) [16]
All Properties
See Table 3.5.2-2
Material Properties (Vapor) [16]
Density
0.5542 kg/m3
Viscosity
1.34E-05 kg/m-s
Thermal Conductivity
0.0261 W/m-K
Surface Tension
0.072 N/m
Solution Methods
Scheme
Coupled
Gradient
Least Square Cell Based
Momentum
Second Order Upwind
Volume Fraction
QUICK
Turbulent Kinetic Energy
Second Order Upwind
Turbulent Dissipation Rate
Second Order Upwind
Energy
Second Order Upwind
* Calculated using equations from Table 2.5.1-2.
Table 3.5.2-2: Subcooled Flow Boiling Model Fluid Properties
Density (kg/m3)
Specific Heat (J/kg-K)
Viscosity (kg/m-s)
Conductivity (W/m-K)
Heat of Vaporization (J/kgmol)
Surface Tension (N/m)
*
368 K
961.99
4210.0
0.0002978
0.6773
-----
370 K
960.59
4212.1
0.0002914
0.6780
-----
373.15 K*
958.46
4215.5
0.0002822
0.6790
40622346
0.0589
Saturation temperature at atmospheric pressure (14.7 psia).
Boiling Model Study
The impact that each boiling model has on liquid volume fraction was
investigated by analyzing a set of cases that implemented the inputs listed in
Table 3.5.2-1 and Table 3.5.2-3. Based on the modeling options available in Fluent, six
combinations were possible. The liquid volume fraction at different axial locations and
the values of average liquid volume fraction among cases were compared.
72
Table 3.5.2-3: Boiling Model Study Case Input
Case
Number
1
2
3
4
5
6
Bubble Departure
Diameter Model
Tolubinski-Kostanchuk
Kocamustafaogullari-Ishii
Unal
Tolubinski-Kostanchuk
Kocamustafaogullari-Ishii
Unal
Nucleation Site
Density Model
Lemmert-Chawla
Lemmert-Chawla
Lemmert-Chawla
Kocamustafaogullari-Ishii
Kocamustafaogullari-Ishii
Kocamustafaogullari-Ishii
Bubble Departure
Frequency Model
Cole
Cole
Cole
Cole
Cole
Cole
Plots of temperature, liquid volume fraction and mass transfer rate for Case 1 are
shown in Figures 3.5.2-2, 3.5.2-3 and 3.5.2-4, respectively. Although these figures are
specific to Case 1, their trends can be applied to all of the subcooled flow boiling cases
analyzed. Figure 3.5.2-2 displays how the liquid temperature increases as the fluid
travels through the pipe. The maximum bulk liquid temperature is about 373 K which is
also the fluid saturation temperature.
Figure 3.5.2-2: Case 1 - Temperature (K)
Figure 3.5.2-3 reveals how the liquid volume fraction decreases as the fluid
travels through the pipe. The large reduction in liquid volume fraction at the pipe exit is
caused by energy transfer from the walls and the small amount of liquid subcooling at
the pipe entrance.
Figure 3.5.2-3: Case 1 - Liquid Volume Fraction
73
Figure 3.5.2-4 is of particular interest because it shows both the generation and
destruction of vapor bubbles. The light blue and teal area next to the heated wall
illustrates that vapor is being generated. After the bubbles grow in size they detach and
join the bulk fluid. A small distance towards the pipe centerline, away from the heated
wall, is a dark blue region.
In this region, the vapor bubbles lose energy to the
surrounding subcooled liquid and condense back into liquid.
The generation and
destruction of vapor bubbles is characteristic of subcooled flow boiling.
(a) Entire Pipe Length
(b) Enhanced View of Pipe Exit
Figure 3.5.2-4: Case 1 - Mass Transfer Rate (kg/m3-s)
The generation and destruction of vapor bubbles can more clearly be seen in
Figure 3.5.2-5. The x-axis represents the distance from the pipe centerline (0.00 m) to
the pipe wall (0.015 m). The y-axis represents the mass transfer rate where positive
values indicate vapor generation (evaporation) and negative values indicate vapor
destruction (condensation). The black line is the mass transfer rate at the pipe inlet and
reveals that a small amount of vapor is produced at the pipe wall. There is no vapor
destruction at this location since the vapor has not had a chance to detach from the wall
and enter the subcooled bulk fluid. The red line represents the mass transfer rate at the
pipe midpoint (0.25 m from the inlet). It too shows that vapor is created on the pipe wall
but at a much greater rate than at the pipe inlet. Between 0.010 m and 0.014 m from the
pipe centerline, the mass transfer rate is negative; therefore, vapor is condensing back to
liquid. The maximum condensation rate of the three locations plotted occurs at the
midpoint of the pipe around 0.0125 m from the pipe centerline. More condensation
74
occurs here than anywhere else because there is a large amount of vapor available to
condense and the bulk fluid remains subcooled enough to absorb energy.
The
condensation rate decreases to zero about 0.010 m from the pipe centerline because all of
the generated vapor has condensed back to liquid at this point. The green line shows the
mass transfer rate at the outlet of the pipe. The greatest amount of vapor production
occurs on the pipe wall at this location. Vapor is also produced (mass transfer rate is
positive) up to 0.003 m from the pipe wall (0.012 m from the pipe centerline). This is
due to localized superheating which can induce a phase change. Note that, the mass
transfer rate is negative at the pipe centerline. This indicates that the bulk fluid remains
subcooled and that vapor production is larger than vapor destruction at this distance from
the pipe inlet.
Figure 3.5.2-5: Case 1 - Vapor Generation Rate
The volume-weighted average liquid volume fraction of the entire control
volume for the six cases described in Table 3.5.2-3 is listed in Table 3.5.2-4. Case 4
predicted the largest liquid volume fraction while Case 2 predicted the smallest liquid
volume fraction; however, the difference between the two cases is only about 0.016.
Therefore, the choice of boiling model seems to have only a small impact on the overall
liquid volume fraction for the conditions examined. The results also show that the
Kocamustafaogullari-Ishii nucleation site density model tends to predict a greater liquid
75
volume fraction, meaning less vapor production, than the Lemmert-Chawla nucleation
site density model. Cases 4 through 6 have a smaller liquid volume fraction range
(0.9124 to 0.9165) than Cases 1 through 3 (0.9003 to 0.9108). This means that when the
Kocamustafaogullari-Ishii nucleation site density model is employed, the choice of the
bubble departure diameter model has less of an impact on liquid volume fraction than if
the Lemmert-Chawla nucleation site density model is employed.
Analyzing the results of the six cases from a bubble departure diameter model
perspective (comparing Cases 1 and 4 to Cases 2 and 5 to Cases 3 and 6), reveals that
there is no tendency for any of the three models examined to have a bias (i.e.,
consistently predict a larger or smaller liquid volume fraction). Thus, the nucleation site
density model has a greater impact on liquid volume fraction than the bubble departure
diameter model.
Table 3.5.2-4: Boiling Model Study Case Results
Case Number
1
2
3
4
5
6
Volume-Weighted Liquid Volume Fraction
0.91078539
0.90031346
0.90856631
0.91649488
0.91612881
0.91241595
Figure 3.5.2-6 shows the liquid volume fraction at nine axial locations as a
function of distance from the pipe center for the six cases described in Table 3.5.2-3.
The x-axis is position, or distance from the centerline, and the pipe wall is located at
0.015 m. Although Table 3.5.2-4 indicates that the models predict similar liquid volume
fractions within the entire control volume, Figure 3.5.2-6 illustrates that there are
noticeable differences between the cases.
First, there is significantly higher liquid
volume fraction near the pipe inlet (0 to 10 cm) in Cases 4 through 6 compared to
Cases 1 through 3. Therefore, vapor formation using the Kocamustafaogullari-Ishii
nucleation site density model requires more energy addition. Second, the liquid volume
fraction 0.008 m from the pipe centerline is significantly less in Cases 1 through 3 than
in Cases 4 through 6. This is due to the smaller vapor production rate at the pipe wall in
Cases 1 through 3.
76
(a) Case 1
(b) Case 2
(c) Case 3
(d) Case 4
(e) Case 5
(f) Case 6
Figure 3.5.2-6: Liquid Volume Faction for Cases 1-6
77
Inlet Conditions Study
A second parametric study using the subcooled boiling model described in
Table 3.5.2-1 was used to determine how fluid temperature, mass flow and heat flux
impact liquid volume fraction. Six additional cases were analyzed in total as part of this
parametric study. For this set of cases, the active nucleation site density is determined
using the Lemmert and Chwala correlation, the bubble departure diameter is determined
using the Tolubinsky and Kostanchuk correlation and the bubble departure frequency is
determined using the Cole correlation. Case 1 from the boiling model study is used as
the nominal case to which the other six cases are compared. Cases 7 through 12 increase
or decrease the heat flux, the fluid temperature or the mass flow rate relative to the
Case 1 value. The inlet conditions for the cases analyzed are listed in Table 3.5.2-5.
Table 3.5.2-5: Inlet Condition Study Case Input
Case Number
1 (base)
7
8
9
10
11
12
Fluid Temperature
(K)
370
370
370
372
368
370
370
Mass Flow
(kg/s)
0.30
0.30
0.30
0.30
0.30
0.33
0.27
Heat Flux
(kW/m2)
90
100
80
90
90
90
90
The volume-weighted average liquid volume fraction of the entire control
volume for the seven cases described in Table 3.5.2-5 is displayed in Table 3.5.2-6.
Table 3.5.2-6 shows that the minimum and maximum liquid volume fractions occur in
Case 9 and Case 10 (fluid temperature variation cases), respectively. The significant
impact that fluid temperature has on liquid volume fraction can be attributed to the large
specific heat of water (4212 J/kg-K). If the specific heat was smaller, the difference in
liquid volume fraction between these two cases and the base case would be less severe.
Comparing the three cases that cause a decrease in liquid volume fraction from the base
case (Cases 7, 9 and 12) to the three cases that cause an increase in liquid volume
fraction from the base case (Cases 8, 10 and 11) demonstrates that the liquid volume
fraction decreases more than it increases for the same delta change in inlet conditions.
78
Thus, changes in inlet conditions near the saturation point will have a larger impact on
liquid volume fraction than changes in inlet conditions farther away from the saturation
point.
Table 3.5.2-6: Inlet Condition Study Case Results
Case Number
1
7
8
9
10
11
12
Volume-Weighted Liquid Volume Fraction
0.91078539
0.87799626
0.93408281
0.57124303
0.96969908
0.92067945
0.89072032
Table 3.5.2-7 shows the liquid volume fraction at nine axial locations for the
cases described in Table 3.5.2-5. This table allows for a finer comparison of the liquid
volume fraction between the cases. Table 3.5.2-7 does not show any irregular trends in
liquid volume fraction and the same relationships between inlet conditions and liquid
volume fraction developed using Table 3.5.2-6 can be drawn using Table 3.5.2-7.
Therefore, making observations based on overall liquid volume fraction is acceptable but
not conclusive.
Table 3.5.2-7: Axial Liquid Volume Fraction
Location*
0 cm
5 cm
10 cm
15 cm
20 cm
25 cm
30 cm
35 cm
40 cm
*
Case 1
Case 7
Case 8
1.00000 1.00000 1.00000
0.99168 0.98880 0.99397
0.97680 0.97050 0.98231
0.96151 0.95036 0.97201
0.93987 0.92220 0.95598
0.91595 0.89812 0.93589
0.89784 0.87830 0.91644
0.88190 0.85540 0.90250
0.85984 0.80840 0.89019
Distance from the pipe inlet.
79
Case 9
1.00000
0.95129
0.87624
0.80165
0.71266
0.57694
0.42719
0.31823
0.25132
Case 10
1.00000
0.99785
0.99390
0.98885
0.98427
0.97895
0.96784
0.95471
0.93927
Case 11
1.00000
0.99348
0.97938
0.96537
0.94748
0.92264
0.90098
0.88448
0.86812
Case 12
1.00000
0.98907
0.97482
0.95578
0.93222
0.91180
0.89572
0.87680
0.83296
(a) Case 7
(b) Case 8
(c) Case 9
(d) Case 10
(e) Case 11
(f) Case 12
Figure 3.5.2-7: Liquid Volume Faction for Cases 7-12
80
Figure 3.5.2-7 illustrates the liquid volume fraction at the different axial locations
in Table 3.5.2-6. The x-axis is position, or distance from the centerline, and the pipe
wall is located at 0.015 m. The impact that fluid temperature (Case 9 and Case 10) has
on liquid volume fraction is extremely visible in Figure 3.5.2-7.
Case 9 shows
significant voiding in the centerline after 25 cm from the pipe inlet due to the high fluid
temperature (subcooling of about 1 K). Case 10 reveals the opposite where 40 cm from
the pipe inlet there is no voiding even at 0.010 m from the pipe centerline.
The liquid volume fraction at the nine axial locations from Cases 7 through 12
were compared to the liquid volume fraction of the base case (Case 1) using the
following three equations for heat flux, fluid temperature and mass flow, respectively,
where i stands for the axial location.
∆(𝑉𝑜𝑖𝑑 𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛) 𝐶𝑎𝑠𝑒 𝑉𝑜𝑖𝑑 𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑖 − 𝐵𝑎𝑠𝑒 𝑉𝑜𝑖𝑑 𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑖
=
∆(𝐻𝑒𝑎𝑡 𝐹𝑙𝑢𝑥)
𝐶𝑎𝑠𝑒 𝐻𝑒𝑎𝑡 𝐹𝑙𝑢𝑥𝑖 − 𝐵𝑎𝑠𝑒 𝐻𝑒𝑎𝑡 𝐹𝑙𝑢𝑥𝑖
∆(𝑉𝑜𝑖𝑑 𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛)
𝐶𝑎𝑠𝑒 𝑉𝑜𝑖𝑑 𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑖 − 𝐵𝑎𝑠𝑒 𝑉𝑜𝑖𝑑 𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑖
=
∆(𝐹𝑙𝑢𝑖𝑑 𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒) 𝐶𝑎𝑠𝑒 𝐹𝑙𝑢𝑖𝑑 𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒𝑖 − 𝐵𝑎𝑠𝑒 𝐹𝑙𝑢𝑖𝑑 𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒𝑖
∆(𝑉𝑜𝑖𝑑 𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛) 𝐶𝑎𝑠𝑒 𝑉𝑜𝑖𝑑 𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑖 − 𝐵𝑎𝑠𝑒 𝑉𝑜𝑖𝑑 𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑖
=
∆(𝑀𝑎𝑠𝑠 𝐹𝑙𝑜𝑤)
𝐶𝑎𝑠𝑒 𝑀𝑎𝑠𝑠 𝐹𝑙𝑜𝑤𝑖 − 𝐵𝑎𝑠𝑒 𝑀𝑎𝑠𝑠 𝐹𝑙𝑜𝑤𝑖
The results of comparing the values from Table 3.5.2-7 using the three above
equations are shown in Table 3.5.2-8. For example, at an axial height of 10 cm, by
increasing the heat flux from 90 kW/m2 to 100 kW/m2 (Case 1 to Case 7) the liquid
volume fraction decreased by 0.0063 or 0.00063 per kW/m2. Similar calculations were
carried out for the remaining axial locations and inlet conditions. The change in liquid
volume fraction at every axial location was averaged to produce an overall impact that
each inlet conditions has on liquid volume fraction.
81
Table 3.5.2-8: Absolute Impact on Liquid Volume Fraction
Height
0 cm
5 cm
10 cm
15 cm
20 cm
25 cm
30 cm
35 cm
40 cm
Average
Case 7
Case 8
0.00000
0.00000
-0.00029
-0.00023
-0.00063
-0.00055
-0.00112
-0.00105
-0.00177
-0.00161
-0.00178
-0.00199
-0.00195
-0.00186
-0.00265
-0.00206
-0.00514
-0.00304
-0.00154 (kW/m2)-1
Case 9
Case 10
0.00000
0.00000
-0.02020
-0.00309
-0.05028
-0.00855
-0.07993
-0.01367
-0.11361
-0.02220
-0.16951
-0.03150
-0.23533
-0.03500
-0.28184
-0.03641
-0.30426
-0.03972
-0.08028 (K)-1
Case 11
Case 12
0.00000
0.00000
0.06000
0.08700
0.08600
0.06600
0.12867
0.19100
0.25367
0.25500
0.22300
0.13833
0.10467
0.07067
0.08600
0.17000
0.27600
0.89600
0.17178 (kg/s)-1
Table 3.5.2-8 reveals the average impact that changing the heat flux, fluid
temperature and mass flow rate has on the liquid volume fraction. Evaluating which of
the three inputs has more impact on liquid volume fraction is difficult to do in absolute
terms (a 1 kg/s increase in mass flow rate is a larger percentage increase than a
10 kW/m2 increase in heat flux). Therefore, the values in Table 3.5.2-8 were compared
on a percentage basis to provide further insight. Table 3.5.2-9 shows the liquid volume
fraction change expected for a 1% change in each inlet condition. The second column of
Table 3.5.2-9 repeats the inlet conditions used in Case 1 (from Table 3.5.2-1), the third
column calculates 1% of the Case 1 input value (for example, 90 kW/m2 * 0.01 =
0.9 kW/m2), the fourth column repeats the results from Table 3.5.2-8, and the fifth
column shows the outcome when columns three and four are multiplied together.
Table 3.5.2-9: Relative Impact on Liquid Volume Fraction
Inlet
Condition
Heat Flux
Temperature
Mass Flow
Case 1
Input
90 kW/m2
370 K
0.3 kg/s
1% of Case 1
Table 3.5.2-8
Input
Results
2
0.9 kW/m
-0.00154 (kW/m2)-1
3.70 K
-0.08028 (K)-1
0.003 kg/s
0.17178 (kg/s)-1
Equivalent Liquid
Volume Fraction
-0.00139
-0.29704
0.00052
Table 3.5.2-9 states that a 1% increase in heat flux causes the average liquid void
fraction to decrease by 0.00139, a 1% increase in temperature causes the average liquid
void fraction to decrease by 0.29704 and a 1% increase in mass flow rate causes the
average liquid void fraction to increase by 0.00052. It is understood that a 1% increase
82
in the fluid temperature from the Case 1 condition would be greater than the saturation
temperature at atmospheric pressure and therefore impossible; however, this exercise
was performed to show how changes in inlet conditions impact liquid void fraction in a
more revealing manner. Table 3.5.2-9 indicates that fluid temperature has the greatest
impact on liquid volume fraction, the wall heat flux has the second greatest impact and
mass flow rate has the smallest impact.
To ensure that the mesh had no significant effect on the results, a mesh validation
was performed. The mesh validation compared the results displayed in this section
(“Mesh 1” in Table 3.5.2-10) to a second mesh with an increased number of finite
volumes (“Mesh 2” in Table 3.5.2-10). The results from the mesh validation shown in
Table 3.5.2-10 prove that the results are mesh independent.
Table 3.5.2-10: Mesh Validation for Subcooled Flow Boiling Model
Number of Nodes
Number of Elements
Max Liquid Velocity (m/s)
Max Gas Velocity (m/s)
Max Phase Transfer (kg/m3-s)
Min Liquid Volume Fraction
Mesh 1
25000
23976
0.8181624
0.9972627
24.87638
0.4876771
83
Mesh 2
31000
29970
0.8199201
0.9982293
26.22442
0.4853158
Difference
24.00 %
25.00 %
0.21 %
0.10 %
5.42 %
-0.48 %
4. DISUSSION AND CONCLUSIONS
This thesis provided theoretical background and development of computational
fluid dynamic models for various fluid flow and heat transfer phenomena including
natural convection, laminar flow, turbulent flow with and without heat transfer, twophase flow, pool boiling and subcooled flow boiling.
Natural convection models of a heated horizontal cylinder and a heated vertical
plate were presented in Section 3.1.
These models implemented the Boussinesq
approximation to calculate temperature induced density gradients and buoyancy forces.
The heated horizontal cylinder model predicted a greater maximum velocity compared to
the heated vertical plate even though the two models used the same surface and bulk
fluid temperatures. The heated vertical plate had a lower maximum velocity due to drag
forces invoked by the heated surface. Both natural convection models showed good
agreement qualitatively and quantitatively with experimental data.
Laminar flow within a pipe was investigated in Section 3.2. The parabolic
velocity profile that is characteristic of laminar flow matched well qualitatively with
experimental data. Also, the radial velocity for most of the pipe was near zero and was
less than 10-3 times the average axial velocity.
Two models involving turbulent flow within a pipe were created as part of
Section 3.3. As expected, the velocity profiles calculated where flat and the velocity
magnitude didn’t decrease until very close to the pipe wall which matched well
qualitatively with experimental data. The wall shear stress reached a maximum at a
short distance from the pipe inlet due to entrance effects causing a surge in radial
velocity which led to a dramatic reduction in axial velocity. The turbulent flow model
with the heat addition was compared to the turbulent flow model without heat addition
and it was determined that there was a small increase in the fluid velocity magnitude for
the scenario with heat addition. The velocity increase was due to the constant mass flow
rate boundary condition at the inlet and the reduction in density caused by energy
addition.
Two-phase flow involving water and air was examined as part of Section 3.4.
The first model was a mixing tank that used a gas jet to stir the liquid. Effects of
84
Rayleigh instability were observed. Before the jet broke the surface of the water, it
became wavy and began to separate into smaller packets with the same volume but less
surface area due to perturbations in the jet that grow over time. The second model
created was a bubble column reactor. Wall-peaking bubbly flow was observed to occur.
Gas holdup due to phase drag forces and displacement was noted. The amount of gas
holdup was found to be related to inlet gas velocity however the relationship was not
linear. A population balance model was employed for two bubble column cases. The
model predicted that the gas bubbles would coalesce and grow in size as they traveled up
the bubble column due to surface tension. When the surface tension was reduced, the
number of bubbles that grew in size dramatically decreased.
Section 3.5 discussed phase transformation due to heat addition in both stagnant
and flowing liquids. The pool boiling model showed the progression of vapor formation
on the heated surface, detachment and liquid refill. Drag forces between the two phases
caused the liquid to travel upwards with the rising vapor but at a slower rate due to slip.
The second phase transformation model developed and the focus of this research
was a subcooled flow boiling model. The impact that different boiling model options
have on liquid volume fraction was investigated. Three bubble departure diameter
models and two nucleation site density models were analyzed using the same inlet
conditions. The bubble departure diameter models did not show any relationship with
liquid volume fraction; however, the Kocamustafaogullari-Ishii nucleation site density
model tended to predict a greater liquid volume fraction, meaning less vapor production,
than the Lemmert-Chawla nucleation site density model.
A second study on how inlet conditions impact the liquid volume fraction during
subcooled flow boiling was explored.
The inlet conditions of heat flux, fluid
temperature and mass flow rate were increased or decreased relative to a base case value.
The difference in liquid volume fraction between scenarios was compared and
relationships relating the inlet conditions with respect to liquid volume fraction were
developed. Overall, the fluid temperature had the greatest impact on liquid volume
fraction, the wall heat flux had the second greatest impact and mass flow rate had the
smallest impact.
85
REFERENCES
1.
Yeoh, G. H; Tu, J. Y., “Modelling Subcooled Boiling Flows,” Nova Science
Publishers, Inc., 2009.
2.
Krepper, E.; Koncar, B.; Egorov, Y., “CFD Modeling of Subcooled Boiling –
Concept,
Validation
and
Application
to
Fuel
Assembly
Design,”
Elsevier B.V., 2006.
3.
Bird, R. B.; Steward, W. E.; Lightfoot, E. N., “Transport Phenomenon,” Wiley &
Sons Inc., 2nd Edition, 2007.
4.
Patankar, S. V., “Numerical Heat Transfer and Fluid Flow,” Hemisphere
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